## 
## Attaching package: 'Matrix'
## The following objects are masked from 'package:tidyr':
## 
##     expand, pack, unpack

1 Context

1.1 Overall Factor Analysis Steps

  1. Are the data suitable for factor analysis?
    • Bartlett’s Sphericity Test Tests whether correlations are different than zero across all the data. If they are close to zero, there owuld be no common factors to find
    • Kaiser-Meyer-Olkin Criterion Test sampling adequacy, measures the degree to wich each variable predicts all other variables in the data set.
  2. Imputation of Missing Variables
    • Generate categorical values for missing data using Multiple Correspondence Analysis/Missing Fuzzy Average Method
  3. Calculate between-item Correlations
    • the polychoric (for orderd categorical variables) and regular Pearson correlation are calculated for later use in factor analysis and for checking other aspects of the factor analysis. The polychoric correlation is the input to the actual factor analysis.
  4. Estimate Number of Factors
    • Parallel analysis
    • Minimim Average Partial (MAP)
  5. Run factor analysis for a range of possible number of factors
    • produces
      • measures of model fit
      • communalities (how much each item covaries with the other items)
      • loadings (correlation between item and factor(s))
  6. Evaluate Reliability of Model..
    • alpha
    • omega
  7. Select best model (number of factors) given model fit statistics and judgement based on understanding of behavior.

1.2 Context Text Description

All data were used in the factorAnalysis1.1.html document you already received.

A series of factoro analyses were run on all items in the file factorAnalysis1.1.html. THe literature suggests doing this so you can compare both statistical measures of how well it fits, and also how logical the grouping of factors is according to knowledge of the subject.

This set of analyses showed that some items had really low communalities, meaning that they didn’t covary much with any of the other items. This means they will not be robust parts of any factor, and they recommend that you remove these from the analysis.

Models with different numbers of factors had different numbers of items with communalities less than 0.4. So a new data set was made for each number of factors, with only the items with communalities of 0.4 or larger.

Similarly, the loading represents the amount of variation in an item due to the factor in which it is classified. Several items had low loadings (most of these items also had low communalities). We removed items with loadings less than 0.32.

Because each number of factors require a different data set (different numbers of items), all the steps had to be rerun sstarting with imputation. Those are the models included in this document.

2 Factor Analyses

Testing analyses from 12 - 18 factors.

2.1 12 Factors

2.1.1 Imputation of Missing Items (12 factors)

include reference on why imputation is better than removing observations, pairwise removing is biased, using means isn’t good.

Categorical missing values using multiple Correspondence Analysis (also called Missing Fuzzy Average method) Josseet al (201 0)

2.1.2 12-Factors: Pre-Imputation Tests for Data Suitability for Factor Analysis

2.1.2.1 Bartlett’s Test of Sphericity

This function tests whether a correlation matrix is significantly different from an identity matrix (Bartlett, 1951). If the Bartlett’s test is not significant, the correlation matrix is not suitable for factor analysis because the variables show too little covariance.

## Warning in polychoric(ddNum12, smooth = TRUE, correct = 0.01): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 916 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.3855482^{4} with 1035 DF, p = 0e+00.

2.1.2.2 Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8510849

2.1.3 12-Factor: Post-Imputation Tests for Data Suitability for Factor Analysis

2.1.3.1 Bartlett’s Test of Sphericity

## Warning in polychoric(imputedNumericDF12, smooth = TRUE, correct = 0.01): The
## items do not have an equal number of response alternatives, global set to
## FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 916 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.2708769^{4} with 1035 DF, p = 0e+00.

2.1.3.2 12-Factor: Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8510849

2.1.4 Between-Item Correlations

2.1.4.1 Pre-Imputation

For factor analysis, it is recommended that some of the item correlationsshould be between 0.3 and 0.9. Polychoric correlations were computed using the polychoric function in the psych package in R with options smooth = TRUE to adjust for non-positive-definite matrices and correct = 0.01 to avoid computation problems with covariance of zero.

The minimum correlation in this data set is -0.477. The maximum correlation in this data set is 0.856.

#### Post-Imputation

The post imputation polyserial correlation was also computed using the polychoric function, with no setting smooth = TRUE and correct = 0.01.

The minimum correlation was -0.467. The maximum correlation was 0.851.

2.1.5 Estimating Number of Factors

2.1.5.1 Parallel method

From the EFAtools documentation:

Various methods for performing parallel analysis. This function uses future_lapply for which a parallel processing plan can be selected. To do so, call library(future) and, for example, plan(multisession); see examples.

Settings Used:
- n.obs = 1117 - eigen_type = “EFA” - fa = “fa” (factor analysis not PCA) - fm = “wls” (weighted least squares because pa gave errors and weighted recommended for ordinal data. - use = “all.obs” (since we have imputed missing values, we can use all data points) - cor = “poly” (use polychoric correlation matrix) - n.iter = 100 (run for 100 iterations)

The suggested number of factors is 11.

2.1.5.2 Minimum Average Partial

MAP is recommended as a way to find the number of factors when the items are ordinal. The lowest value indicates the best number of factors.

Number of Factors MAP value
1 0.0339
2 0.0297
3 0.0230
4 0.0200
5 0.0181
6 0.0174
7 0.0164
8 0.0158
9 0.0140
10 0.0141
11 0.0142
12 0.0148
13 0.0156
14 0.0165
15 0.0176
16 0.0188
17 0.0204
18 0.0214

2.1.6 Factor Analysis for 12 Factors

2.1.6.1 12 Factors Model Fit

## Warning in polychoric(r, correct = correct, weight = weight): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 916 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

Although the chi-square test of goodness of fit is sensitive to departures from normality like the C-BARQ items, Hopper et al (2008) recommend always reporting it.

  • chi-square: 4483.8490673
  • degrees of freedom: 549
  • P-value for chi-square = 0e+00

Tucker-Lewis Index of Factoring Reliability/Non-Norm Fit Index: 0.7639861. Should be > 0.9; need reference)

2.1.6.2 12 Factor Model Communalities

fanal12.communality
TRAIN01 0.46
TRAIN03 0.43
TRAIN04 0.59
TRAIN07 0.62
AGG09 0.78
AGG10 0.67
AGG14 0.48
AGG15 0.76
AGG16 0.76
AGG18 0.72
AGG19 0.73
AGG20 0.52
FEAR21 0.83
FEAR23 0.80
FEAR24 0.86
FEAR25 0.42
FEAR26 0.42
FEAR27 0.81
FEAR28 0.66
FEAR30 0.35
FEAR31 0.71
SEPR32 0.68
SEPR33 0.66
SEPR34 0.65
EXCITE35 0.46
EXCITE36 0.57
EXCITE37 0.64
EXCITE38 0.64
PLAY43 0.55
PLAY44 0.82
PLAY45 0.77
PLAY46 0.72
IMP47 0.58
IMP48 0.47
MISC52 0.56
MISC53 0.51
MISC54 0.65
MISC55 0.68
MISC56 0.62
MISC59 0.61
MISC61 0.47
MISC62 0.43
MISC64 0.60
MISC65 0.65
MISC66 0.69
MISC67 0.59

2.1.6.3 How many communalities < 0.40?

There are 1 items with communality < 0.40.

fanal12.communality item
FEAR30 0.35 FEAR30

2.1.6.4 12 Factor Model Loadings

PA1 PA3 PA11 PA2 PA7 PA6 PA5 PA8 PA4 PA10 PA12 PA9
TRAIN01 -0.09 0.12 0.05 -0.04 -0.08 -0.05 0.03 0.00 -0.01 0.59 0.02 0.06
TRAIN03 -0.02 -0.03 0.00 -0.02 0.03 0.08 0.11 -0.04 0.03 0.61 0.03 0.02
TRAIN04 0.04 0.20 -0.02 -0.05 -0.01 0.02 0.02 0.01 -0.07 0.63 0.11 -0.05
TRAIN07 0.06 0.60 -0.08 -0.06 -0.05 0.00 0.02 0.01 -0.01 0.30 -0.02 -0.02
AGG09 0.19 0.03 0.77 0.03 0.08 -0.03 -0.05 0.00 -0.03 0.05 0.08 -0.10
AGG10 -0.03 -0.02 0.80 0.09 -0.02 -0.01 0.12 0.01 0.06 -0.03 -0.06 0.06
AGG14 0.00 0.08 0.66 0.02 -0.01 0.05 0.11 0.01 0.02 -0.05 -0.05 0.11
AGG15 0.27 0.01 0.73 -0.03 0.02 -0.01 -0.04 -0.11 0.12 0.00 0.01 0.06
AGG16 -0.19 0.05 0.34 -0.06 0.05 0.01 -0.05 0.37 0.43 0.06 0.00 -0.18
AGG18 -0.20 0.01 0.32 -0.02 -0.03 -0.05 -0.03 0.34 0.47 0.03 0.10 -0.14
AGG19 0.02 0.04 0.05 0.05 -0.01 0.09 0.02 0.01 0.80 -0.01 -0.02 -0.07
AGG20 0.05 -0.04 -0.07 -0.07 0.03 -0.03 0.01 -0.07 0.74 -0.02 0.09 0.13
FEAR21 0.81 0.00 0.14 -0.03 0.02 0.01 0.00 0.09 0.01 0.00 0.04 -0.03
FEAR23 0.70 0.00 0.20 -0.06 0.04 0.00 0.05 0.17 -0.06 -0.03 0.06 -0.12
FEAR24 0.79 0.02 0.16 -0.12 -0.04 0.06 -0.02 0.16 -0.02 -0.05 0.02 -0.04
FEAR25 0.32 0.17 -0.07 0.03 0.29 0.03 0.04 0.09 0.06 0.12 -0.01 0.03
FEAR26 0.53 0.03 -0.03 0.12 0.01 -0.05 0.10 0.06 0.24 0.04 -0.05 0.20
FEAR27 0.15 0.00 -0.03 0.01 0.02 -0.01 0.02 0.84 0.03 -0.02 0.00 0.03
FEAR28 0.55 0.11 -0.11 0.08 0.38 0.03 0.05 0.05 0.08 0.05 -0.04 -0.03
FEAR30 0.25 0.06 -0.11 0.10 0.17 -0.06 0.15 0.03 0.29 0.06 -0.06 0.28
FEAR31 0.12 0.02 -0.11 0.01 0.06 0.00 0.04 0.80 -0.04 0.02 -0.01 0.08
SEPR32 0.09 0.00 0.00 -0.07 -0.04 0.07 0.78 0.00 0.06 -0.06 0.04 -0.07
SEPR33 -0.06 0.02 0.00 0.02 0.02 0.00 0.81 0.05 -0.03 0.04 -0.01 0.02
SEPR34 -0.06 -0.01 0.07 -0.01 0.05 -0.03 0.78 0.01 -0.05 0.07 0.04 0.03
EXCITE35 0.08 0.07 0.00 -0.09 -0.11 0.63 0.14 0.01 -0.01 -0.03 0.03 -0.01
EXCITE36 0.05 0.05 -0.02 0.18 0.02 0.67 -0.04 -0.08 0.01 -0.04 0.04 -0.01
EXCITE37 -0.03 0.00 0.03 -0.07 0.00 0.82 0.01 0.02 0.00 0.02 -0.01 0.01
EXCITE38 -0.01 -0.12 -0.04 0.05 0.08 0.76 -0.03 0.02 0.03 0.06 0.00 0.05
PLAY43 -0.26 0.02 -0.26 0.53 0.00 0.08 -0.04 0.08 -0.06 -0.07 0.02 0.05
PLAY44 0.02 0.01 -0.03 0.86 -0.03 0.04 -0.02 0.00 0.10 0.01 0.04 -0.08
PLAY45 0.00 0.01 0.06 0.83 -0.05 0.00 -0.05 0.07 -0.09 -0.12 0.06 0.00
PLAY46 -0.02 -0.15 0.14 0.79 0.01 0.02 0.04 -0.08 -0.02 0.07 -0.02 -0.02
IMP47 0.02 -0.10 0.01 0.12 0.02 0.08 -0.04 0.06 0.00 0.24 0.60 0.12
IMP48 -0.01 -0.01 0.00 0.06 0.05 0.09 0.03 0.05 0.06 0.13 0.51 0.15
MISC52 -0.07 0.09 0.15 -0.09 0.01 0.07 -0.04 0.12 -0.05 -0.04 0.03 0.69
MISC53 -0.08 -0.02 0.03 -0.10 -0.04 0.02 0.03 0.02 0.04 0.09 0.15 0.62
MISC54 0.02 0.14 -0.01 -0.04 0.72 0.06 -0.02 0.02 0.02 0.02 -0.05 0.08
MISC55 -0.03 -0.05 0.03 0.03 0.84 0.01 0.07 0.00 0.02 -0.01 -0.05 -0.07
MISC56 -0.08 0.00 0.05 -0.08 0.79 -0.02 -0.03 0.03 -0.04 -0.04 0.13 0.01
MISC59 0.06 0.12 -0.10 0.07 0.01 0.00 0.27 -0.10 0.08 -0.06 0.60 0.01
MISC61 -0.03 0.09 0.06 0.27 -0.10 0.23 0.16 -0.11 -0.05 -0.16 0.31 -0.15
MISC62 -0.14 0.42 0.07 -0.02 -0.02 0.07 -0.02 0.31 0.11 0.01 0.13 -0.03
MISC64 -0.10 0.78 0.06 0.00 0.00 -0.02 0.03 -0.04 -0.01 0.00 -0.04 0.02
MISC65 -0.09 0.72 0.03 -0.02 -0.01 -0.01 0.02 0.05 0.05 0.11 0.02 0.07
MISC66 0.23 0.59 -0.01 -0.05 0.20 -0.05 0.03 -0.02 -0.04 0.08 0.05 0.01
MISC67 0.18 0.58 0.01 -0.12 0.16 -0.02 0.03 0.02 -0.01 -0.07 0.09 0.00
2.1.6.4.1 12 Factor Model Largest Loading Per Item
Largest Loading Per Item and Associated Factors, Sorted by Factor
largest maximum itemNames itemDescriptions
13 PA1 0.81 FEAR21 When an unfamiliar person approaches the dog when s/he is away from his/her normal home environment or kennel
14 PA1 0.70 FEAR23 When an unfamiliar person visits your home or approaches the dog when in his/her home kennel
15 PA1 0.79 FEAR24 When an unfamiliar person tries to touch or pet the dog
16 PA1 0.32 FEAR25 In response to strange or unfamiliar objects on or near sidewalks or walkways (e.g., plastic trash bags, leaves, litter, flags flapping, etc.
17 PA1 0.53 FEAR26 When examined/treated by a veterinarian
19 PA1 0.55 FEAR28 When first exposed to unfamiliar situations (e.g., novel environments, first visit to the veterinarian, etc.)
43 PA10 0.59 TRAIN01 Is hard to recall when off the leash
44 PA10 0.61 TRAIN03 Is slow to obey a ‘stay’ command
45 PA10 0.63 TRAIN04 Has difficulty attending/listening to things you say or do
1 PA11 0.77 AGG09 Barks, growls, attempts to bite when approached directly by an unfamiliar person while being walked/exercised on a leash
2 PA11 0.80 AGG10 Barks, etc. when unfamiliar persons approach the dog when s/he is in his/her kennel
3 PA11 0.66 AGG14 When strangers walk past when the dog is in his/her home run or kennel
4 PA11 0.73 AGG15 When an unfamiliar person tries to touch or pet the dog
22 PA12 0.60 IMP47 Impulsive; doesn’t seem to think before s/he acts
23 PA12 0.51 IMP48 Becomes frustrated/impatient in a wide range of situations
29 PA12 0.60 MISC59 Hyperactive, restless, has trouble settling down
30 PA12 0.31 MISC61 Active, energetic, always on the go
36 PA2 0.53 PLAY43 Eagerly engages in play with new/unfamiliar people
37 PA2 0.86 PLAY44 Highly toy focused; attention riveted on tug toy/balls when these are held by handler or other person
38 PA2 0.83 PLAY45 Eagerly initiates play sessions; brings objects/toys to you/the handler and retrieves them when thrown
39 PA2 0.79 PLAY46 Hunts persistently for thrown or hidden toys/objects, not easily distracted from this task
31 PA3 0.42 MISC62 Becomes highly excited and/or distracted when encountering unfamiliar dogs
32 PA3 0.78 MISC64 When working, is easily distracted or preoccupied by odors/engages in persistent sniffing of ground or objects
33 PA3 0.72 MISC65 Has difficulty shifting attention away from interesting or distracting stimuli (e.g., other dogs, odor, people, small animals, etc.)
34 PA3 0.59 MISC66 Is distracted or nervous in new, unfamiliar environments, has difficulty maintaining focus on work
35 PA3 0.58 MISC67 Is slow to recover after being distracted, startled, or frightened/takes a long time to resume work
46 PA3 0.60 TRAIN07 Unfocused; is easily distracted by interesting sights, sounds or smells
5 PA4 0.43 AGG16 When approached directly by an unfamiliar dog while being walked/exercised on a leash
6 PA4 0.47 AGG18 When barked, growled, or lunged at by another dog
7 PA4 0.80 AGG19 Towards other familiar dogs
8 PA4 0.74 AGG20 When approached while playing with/chewing a favorite toy, bone, object, etc. by another familiar dog
20 PA4 0.29 FEAR30 When groomed or bathed
40 PA5 0.78 SEPR32 Restlessness/agitation/pacing when left alone
41 PA5 0.81 SEPR33 Barking or whining when left alone
42 PA5 0.78 SEPR34 Chewing/scratching at doors, floor, fencing, etc., when left alone
9 PA6 0.63 EXCITE35 Excitable when you first arrive home, or at the dog’s kennel, after a brief absence
10 PA6 0.67 EXCITE36 Excitable when playing with you or other familiar persons
11 PA6 0.82 EXCITE37 Excitable just before being taken out for a walk
12 PA6 0.76 EXCITE38 Excitable just before being taken out for work or training
26 PA7 0.72 MISC54 Reluctant to/nervous about crossing grates or other unfamiliar surfaces
27 PA7 0.84 MISC55 Reluctant to/nervous about crossing shiny or slippery floors
28 PA7 0.79 MISC56 Nervous or frightened when ascending or descending some types of stairs
18 PA8 0.84 FEAR27 When approached directly by an unfamiliar dog
21 PA8 0.80 FEAR31 When barked, growled, or lunged at by an unfamiliar dog
24 PA9 0.69 MISC52 Begs persistently for food when people are eating
25 PA9 0.62 MISC53 Steals food

2.1.7 12 Model Reliability Measures

PA1 PA3 PA11 PA2 PA7 PA6 PA5 PA8 PA4 PA10 PA12 PA9
alpha 0.797 0.787 0.76 0.812 0.718 0.787 0.735 0.803 0.715 0.652 0.661 0.623

omega is thought to be better measure of reliability than alpha, because it looks at the relationships between all the items rather than just the ones within each factor. It’s different than alpha because it assumes that there is a genral factor, g, that all items have in common. This is listed as the first row of the table below.

There are three omegas for each factor:

  • omega_total - total true score variance in factor
  • omega_hierarchical - variance attributable to general factor (g)
  • omega_subscale - variance attributable to the subscale

From EFAtools::OMEGA documentation:

Omegas refer to the correlation between a factor and a unit-weighted composite score and thus the true score variance in a unit-weighted composite based on the respective indicators. Omega total is the total true score variance in a composite. Omega hierarchical is the true score variance in a composite that is attributable to the general factor, and omega subscale is the true score variance in a composite attributable to all subscales / group factors (for the whole scale) or to the specific subscale / group factor (for subscale composites).

Interpretation of omega_subscale

no universal guideline

0.50 is minimum, 0.75 preferable (watkins books,rodrigues16, zimbarg2005

## Omega total, omega hierarchical, omega subscale, H index, explained common variance (ECV), and percent of uncontaminated correlations (PUC) for the general factor (top row) and omegas and H index for the group factors:
## 
##       tot  hier   sub     H  ECV   PUC
## g   0.943 0.548 0.215 0.875 0.24 0.929
## F1  0.818 0.314 0.504 0.786           
## F2  0.640 0.159 0.482 0.575           
## F3  0.824 0.132 0.691 0.806           
## F4  0.514 0.091 0.422 0.575           
## F5  0.825 0.069 0.756 0.860           
## F6  0.830 0.407 0.424 0.688           
## F7  0.654 0.181 0.473 0.717           
## F8  0.844 0.147 0.697 0.773           
## F9  0.803 0.015 0.787 0.825           
## F10 0.839 0.185 0.654 0.752           
## F11 0.819 0.223 0.596 0.696           
## F12 0.633 0.129 0.504 0.611

2.2 13 Factors

2.2.1 Imputation of Missing Items (13 factors)

include reference on why imputation is better than removing observations, pairwise removing is biased, using means isn’t good.

Categorical missing values using multiple Correspondence Analysis (also called Missing Fuzzy Average method) Josseet al (201 0)

2.2.2 13-Factors: Pre-Imputation Tests for Data Suitability for Factor Analysis

2.2.2.1 Bartlett’s Test of Sphericity

This function tests whether a correlation matrix is significantly different from an identity matrix (Bartlett, 1951). If the Bartlett’s test is not significant, the correlation matrix is not suitable for factor analysis because the variables show too little covariance.

## Warning in polychoric(ddNum13, smooth = TRUE, correct = 0.01): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1005 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.6452156^{4} with 1128 DF, p = 0e+00.

2.2.2.2 Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8506322

2.2.3 13-Factor: Post-Imputation Tests for Data Suitability for Factor Analysis

2.2.3.1 Bartlett’s Test of Sphericity

## Warning in polychoric(imputedNumericDF13, smooth = TRUE, correct = 0.01): The
## items do not have an equal number of response alternatives, global set to
## FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1005 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.505972^{4} with 1128 DF, p = 0e+00.

2.2.3.2 13-Factor: Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8506322

2.2.4 Between-Item Correlations

2.2.4.1 Pre-Imputation

For factor analysis, it is recommended that some of the item correlationsshould be between 0.3 and 0.9. Polychoric correlations were computed using the polychoric function in the psych package in R with options smooth = TRUE to adjust for non-positive-definite matrices and correct = 0.01 to avoid computation problems with covariance of zero.

The minimum correlation in this data set is -0.477. The maximum correlation in this data set is 0.856.

#### Post-Imputation

The post imputation polyserial correlation was also computed using the polychoric function, with no setting smooth = TRUE and correct = 0.01.

The minimum correlation was -0.467. The maximum correlation was 0.851.

2.2.5 Estimating Number of Factors

2.2.5.1 Parallel method

From the EFAtools documentation:

Various methods for performing parallel analysis. This function uses future_lapply for which a parallel processing plan can be selected. To do so, call library(future) and, for example, plan(multisession); see examples.

Settings Used:
- n.obs = 1117 - eigen_type = “EFA” - fa = “fa” (factor analysis not PCA) - fm = “wls” (weighted least squares because pa gave errors and weighted recommended for ordinal data. - use = “all.obs” (since we have imputed missing values, we can use all data points) - cor = “poly” (use polychoric correlation matrix) - n.iter = 100 (run for 100 iterations)

The suggested number of factors is 11.

2.2.5.2 Minimum Average Partial

MAP is recommended as a way to find the number of factors when the items are ordinal. The lowest value indicates the best number of factors.

## Warning in fa.stats(r = r, f = f, phi = phi, n.obs = n.obs, np.obs = np.obs, :
## The estimated weights for the factor scores are probably incorrect.  Try a
## different factor score estimation method.
## Warning in fac(r = r, nfactors = nfactors, n.obs = n.obs, rotate = rotate, : An
## ultra-Heywood case was detected.  Examine the results carefully
## Warning in fa.stats(r = r, f = f, phi = phi, n.obs = n.obs, np.obs = np.obs, :
## The estimated weights for the factor scores are probably incorrect.  Try a
## different factor score estimation method.
## Warning in fac(r = r, nfactors = nfactors, n.obs = n.obs, rotate = rotate, : An
## ultra-Heywood case was detected.  Examine the results carefully
## Warning in fa.stats(r = r, f = f, phi = phi, n.obs = n.obs, np.obs = np.obs, :
## The estimated weights for the factor scores are probably incorrect.  Try a
## different factor score estimation method.
## Warning in fac(r = r, nfactors = nfactors, n.obs = n.obs, rotate = rotate, : An
## ultra-Heywood case was detected.  Examine the results carefully
## Warning in fa.stats(r = r, f = f, phi = phi, n.obs = n.obs, np.obs = np.obs, :
## The estimated weights for the factor scores are probably incorrect.  Try a
## different factor score estimation method.
## Warning in fac(r = r, nfactors = nfactors, n.obs = n.obs, rotate = rotate, : An
## ultra-Heywood case was detected.  Examine the results carefully
Number of Factors MAP value
1 0.0326
2 0.0288
3 0.0221
4 0.0195
5 0.0179
6 0.0174
7 0.0168
8 0.0157
9 0.0144
10 0.0145
11 0.0144
12 0.0147
13 0.0154
14 0.0158
15 0.0167
16 0.0177
17 0.0191
18 0.0203

2.2.6 Factor Analysis for 13 Factors

2.2.6.1 13 Factors Model Fit

## Warning in polychoric(r, correct = correct, weight = weight): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1005 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

Although the chi-square test of goodness of fit is sensitive to departures from normality like the C-BARQ items, Hopper et al (2008) recommend always reporting it.

  • chi-square: 5317.5606254
  • degrees of freedom: 582
  • P-value for chi-square = 0e+00

Tucker-Lewis Index of Factoring Reliability/Non-Norm Fit Index: 0.7273123. Should be > 0.9; need reference)

2.2.6.2 13 Factor Model Communalities

fanal13.communality
TRAIN01 0.47
TRAIN03 0.45
TRAIN04 0.59
TRAIN07 0.62
AGG09 0.78
AGG10 0.67
AGG14 0.48
AGG15 0.79
AGG16 0.78
AGG17 0.76
AGG18 0.73
AGG19 0.68
AGG20 0.65
FEAR21 0.83
FEAR23 0.81
FEAR24 0.87
FEAR25 0.44
FEAR26 0.44
FEAR27 0.81
FEAR28 0.65
FEAR29 0.53
FEAR30 0.67
FEAR31 0.71
SEPR32 0.67
SEPR33 0.63
SEPR34 0.70
EXCITE35 0.47
EXCITE36 0.59
EXCITE37 0.63
EXCITE38 0.65
PLAY43 0.56
PLAY44 0.80
PLAY45 0.76
PLAY46 0.71
IMP47 0.51
IMP48 0.43
MISC52 0.52
MISC53 0.50
MISC54 0.66
MISC55 0.69
MISC56 0.60
MISC59 0.52
MISC61 0.51
MISC62 0.44
MISC64 0.65
MISC65 0.65
MISC66 0.69
MISC67 0.60

2.2.6.3 How many communalities < 0.40?

There are 0 items with communality < 0.40.

fanal13.communality item

2.2.6.4 13 Factor Model Loadings

PA1 PA12 PA11 PA2 PA7 PA8 PA9 PA5 PA4 PA3 PA10 PA6 PA13
TRAIN01 -0.14 0.05 0.12 -0.08 -0.08 -0.04 0.01 0.02 -0.05 0.60 0.07 0.03 0.09
TRAIN03 -0.07 0.03 -0.07 -0.09 0.06 0.08 0.08 -0.05 -0.04 0.62 0.11 -0.03 -0.11
TRAIN04 0.07 -0.06 0.21 -0.04 0.00 0.01 0.02 0.01 -0.03 0.65 -0.03 0.01 0.06
TRAIN07 0.05 -0.10 0.59 -0.10 -0.05 0.00 0.01 0.02 -0.01 0.29 0.03 -0.05 0.02
AGG09 0.22 0.79 0.02 0.05 0.08 -0.04 -0.04 -0.02 -0.05 0.08 0.04 -0.06 -0.04
AGG10 -0.04 0.78 -0.02 0.06 -0.03 -0.01 0.11 0.03 0.03 -0.06 0.01 0.05 0.12
AGG14 0.04 0.59 0.09 -0.01 -0.01 0.03 0.11 0.03 0.07 -0.09 -0.09 0.16 0.15
AGG15 0.26 0.75 0.00 -0.05 0.01 0.00 -0.04 -0.11 0.07 0.00 0.09 0.04 0.04
AGG16 -0.15 0.40 0.03 -0.06 0.08 0.01 -0.05 0.37 0.38 0.08 -0.04 -0.20 -0.14
AGG17 0.06 0.28 0.09 0.12 0.06 0.10 0.13 0.01 0.13 0.01 0.11 -0.15 0.61
AGG18 -0.17 0.38 0.02 0.03 -0.01 -0.04 0.00 0.34 0.43 0.07 -0.02 -0.13 -0.09
AGG19 -0.03 0.11 0.02 0.02 0.00 0.11 0.01 0.04 0.69 -0.02 0.09 -0.16 -0.04
AGG20 0.06 -0.12 -0.01 -0.05 0.01 -0.03 0.01 -0.06 0.84 -0.03 0.05 0.13 0.09
FEAR21 0.80 0.11 0.00 -0.02 0.02 0.01 0.00 0.08 0.02 0.00 0.09 -0.03 0.04
FEAR23 0.76 0.16 0.00 -0.03 0.06 -0.01 0.07 0.15 -0.01 -0.01 -0.04 -0.05 0.00
FEAR24 0.80 0.14 0.01 -0.11 -0.02 0.05 -0.02 0.15 -0.01 -0.04 0.04 -0.02 -0.01
FEAR25 0.33 -0.12 0.17 0.02 0.30 0.02 0.04 0.10 0.10 0.11 -0.03 0.04 0.10
FEAR26 0.36 0.01 0.00 0.05 -0.03 -0.02 0.04 0.07 0.09 0.00 0.42 0.04 0.01
FEAR27 0.18 -0.04 0.01 0.02 0.02 0.00 0.02 0.83 0.04 -0.02 0.01 0.04 0.02
FEAR28 0.47 -0.12 0.10 0.04 0.37 0.05 0.03 0.06 0.02 0.03 0.15 -0.10 0.06
FEAR29 -0.05 0.04 -0.01 0.00 -0.05 0.01 0.01 0.04 -0.03 0.04 0.73 0.02 0.05
FEAR30 -0.03 -0.02 0.01 0.02 0.08 -0.02 0.05 0.00 0.06 -0.01 0.77 0.03 -0.02
FEAR31 0.11 -0.11 0.02 0.02 0.05 0.00 0.03 0.79 -0.07 0.01 0.10 0.06 0.01
SEPR32 0.09 -0.01 0.00 -0.06 -0.04 0.07 0.79 0.00 0.05 -0.06 0.04 -0.08 -0.01
SEPR33 -0.07 -0.02 0.01 -0.03 0.03 -0.01 0.78 0.04 -0.04 0.01 0.06 -0.01 -0.06
SEPR34 -0.07 0.02 0.00 -0.01 0.05 -0.04 0.82 0.03 -0.04 0.07 -0.02 0.04 0.13
EXCITE35 0.10 0.02 0.05 -0.10 -0.09 0.64 0.15 0.00 -0.03 -0.02 -0.01 -0.01 -0.12
EXCITE36 0.01 0.00 0.04 0.18 0.01 0.70 -0.05 -0.09 -0.03 -0.03 0.06 -0.04 -0.01
EXCITE37 -0.02 0.01 0.00 -0.08 -0.01 0.82 0.01 0.02 0.00 0.01 -0.03 0.02 0.02
EXCITE38 -0.02 -0.07 -0.11 0.04 0.07 0.77 -0.04 0.03 0.04 0.05 -0.02 0.06 0.09
PLAY43 -0.28 -0.24 0.00 0.52 0.02 0.08 -0.05 0.07 -0.08 -0.08 0.00 0.03 -0.11
PLAY44 -0.02 -0.03 -0.02 0.85 -0.03 0.05 -0.04 0.00 0.07 0.00 0.04 -0.10 0.00
PLAY45 -0.02 0.06 -0.01 0.85 -0.05 0.01 -0.05 0.06 -0.09 -0.11 -0.01 0.01 0.00
PLAY46 -0.04 0.09 -0.16 0.75 0.01 0.02 0.03 -0.05 -0.02 0.05 -0.05 -0.03 0.12
IMP47 0.12 0.00 -0.03 0.35 0.01 0.09 0.06 0.03 0.14 0.38 -0.05 0.34 -0.05
IMP48 0.08 0.01 0.04 0.25 0.05 0.10 0.11 0.02 0.16 0.25 -0.01 0.33 -0.13
MISC52 -0.11 0.14 0.08 -0.15 0.03 0.08 -0.05 0.14 -0.03 -0.06 0.09 0.64 -0.01
MISC53 -0.12 0.06 -0.03 -0.13 -0.01 0.03 0.04 0.02 0.04 0.10 0.12 0.60 -0.12
MISC54 -0.02 -0.03 0.14 -0.06 0.70 0.06 -0.04 0.03 0.00 0.00 0.07 0.06 0.09
MISC55 -0.03 0.01 -0.07 0.00 0.85 -0.01 0.05 -0.01 0.01 -0.03 -0.01 -0.06 -0.03
MISC56 -0.05 0.06 0.01 -0.02 0.78 -0.02 -0.01 0.01 -0.02 0.01 0.00 0.06 -0.02
MISC59 0.15 -0.05 0.15 0.30 0.02 0.02 0.37 -0.13 0.16 0.11 0.00 0.19 -0.22
MISC61 0.05 0.11 0.08 0.40 -0.07 0.23 0.22 -0.16 -0.02 -0.07 -0.06 -0.04 -0.31
MISC62 -0.11 0.13 0.41 0.02 0.00 0.08 0.01 0.30 0.09 0.06 0.01 -0.02 -0.19
MISC64 -0.11 0.03 0.82 -0.01 -0.02 -0.02 0.03 -0.02 -0.01 -0.03 0.00 0.01 0.11
MISC65 -0.10 0.03 0.71 -0.03 -0.01 -0.01 0.02 0.06 0.03 0.10 0.05 0.04 -0.02
MISC66 0.23 -0.01 0.58 -0.04 0.21 -0.05 0.03 -0.03 -0.05 0.09 0.06 0.01 -0.08
MISC67 0.22 0.00 0.58 -0.08 0.17 -0.03 0.05 0.00 0.03 -0.05 0.01 0.03 -0.08
2.2.6.4.1 13 Factor Model Largest Loading Per Item
Largest Loading Per Item and Associated Factors, Sorted by Factor
largest maximum itemNames itemDescriptions
14 PA1 0.80 FEAR21 When an unfamiliar person approaches the dog when s/he is away from his/her normal home environment or kennel
15 PA1 0.76 FEAR23 When an unfamiliar person visits your home or approaches the dog when in his/her home kennel
16 PA1 0.80 FEAR24 When an unfamiliar person tries to touch or pet the dog
17 PA1 0.33 FEAR25 In response to strange or unfamiliar objects on or near sidewalks or walkways (e.g., plastic trash bags, leaves, litter, flags flapping, etc.
20 PA1 0.47 FEAR28 When first exposed to unfamiliar situations (e.g., novel environments, first visit to the veterinarian, etc.)
18 PA10 0.42 FEAR26 When examined/treated by a veterinarian
21 PA10 0.73 FEAR29 When having nails trimmed, or feet touched/handled
22 PA10 0.77 FEAR30 When groomed or bathed
33 PA11 0.41 MISC62 Becomes highly excited and/or distracted when encountering unfamiliar dogs
34 PA11 0.82 MISC64 When working, is easily distracted or preoccupied by odors/engages in persistent sniffing of ground or objects
35 PA11 0.71 MISC65 Has difficulty shifting attention away from interesting or distracting stimuli (e.g., other dogs, odor, people, small animals, etc.)
36 PA11 0.58 MISC66 Is distracted or nervous in new, unfamiliar environments, has difficulty maintaining focus on work
37 PA11 0.58 MISC67 Is slow to recover after being distracted, startled, or frightened/takes a long time to resume work
48 PA11 0.59 TRAIN07 Unfocused; is easily distracted by interesting sights, sounds or smells
1 PA12 0.79 AGG09 Barks, growls, attempts to bite when approached directly by an unfamiliar person while being walked/exercised on a leash
2 PA12 0.78 AGG10 Barks, etc. when unfamiliar persons approach the dog when s/he is in his/her kennel
3 PA12 0.59 AGG14 When strangers walk past when the dog is in his/her home run or kennel
4 PA12 0.75 AGG15 When an unfamiliar person tries to touch or pet the dog
5 PA12 0.40 AGG16 When approached directly by an unfamiliar dog while being walked/exercised on a leash
6 PA13 0.61 AGG17 When stared at directly by you or another familiar person
32 PA2 0.40 MISC61 Active, energetic, always on the go
38 PA2 0.52 PLAY43 Eagerly engages in play with new/unfamiliar people
39 PA2 0.85 PLAY44 Highly toy focused; attention riveted on tug toy/balls when these are held by handler or other person
40 PA2 0.85 PLAY45 Eagerly initiates play sessions; brings objects/toys to you/the handler and retrieves them when thrown
41 PA2 0.75 PLAY46 Hunts persistently for thrown or hidden toys/objects, not easily distracted from this task
24 PA3 0.38 IMP47 Impulsive; doesn’t seem to think before s/he acts
45 PA3 0.60 TRAIN01 Is hard to recall when off the leash
46 PA3 0.62 TRAIN03 Is slow to obey a ‘stay’ command
47 PA3 0.65 TRAIN04 Has difficulty attending/listening to things you say or do
7 PA4 0.43 AGG18 When barked, growled, or lunged at by another dog
8 PA4 0.69 AGG19 Towards other familiar dogs
9 PA4 0.84 AGG20 When approached while playing with/chewing a favorite toy, bone, object, etc. by another familiar dog
19 PA5 0.83 FEAR27 When approached directly by an unfamiliar dog
23 PA5 0.79 FEAR31 When barked, growled, or lunged at by an unfamiliar dog
25 PA6 0.33 IMP48 Becomes frustrated/impatient in a wide range of situations
26 PA6 0.64 MISC52 Begs persistently for food when people are eating
27 PA6 0.60 MISC53 Steals food
28 PA7 0.70 MISC54 Reluctant to/nervous about crossing grates or other unfamiliar surfaces
29 PA7 0.85 MISC55 Reluctant to/nervous about crossing shiny or slippery floors
30 PA7 0.78 MISC56 Nervous or frightened when ascending or descending some types of stairs
10 PA8 0.64 EXCITE35 Excitable when you first arrive home, or at the dog’s kennel, after a brief absence
11 PA8 0.70 EXCITE36 Excitable when playing with you or other familiar persons
12 PA8 0.82 EXCITE37 Excitable just before being taken out for a walk
13 PA8 0.77 EXCITE38 Excitable just before being taken out for work or training
31 PA9 0.37 MISC59 Hyperactive, restless, has trouble settling down
42 PA9 0.79 SEPR32 Restlessness/agitation/pacing when left alone
43 PA9 0.78 SEPR33 Barking or whining when left alone
44 PA9 0.82 SEPR34 Chewing/scratching at doors, floor, fencing, etc., when left alone

2.2.7 13 Model Reliability Measures

PA1 PA12 PA11 PA2 PA7 PA8 PA9 PA5 PA4 PA3 PA10 PA6 PA13
alpha 0.817 0.751 0.787 0.8 0.718 0.787 0.718 0.804 0.626 0.618 0.616 0.54 1

omega is thought to be better measure of reliability than alpha, because it looks at the relationships between all the items rather than just the ones within each factor. It’s different than alpha because it assumes that there is a genral factor, g, that all items have in common. This is listed as the first row of the table below.

There are three omegas for each factor:

  • omega_total - total true score variance in factor
  • omega_hierarchical - variance attributable to general factor (g)
  • omega_subscale - variance attributable to the subscale

From EFAtools::OMEGA documentation:

Omegas refer to the correlation between a factor and a unit-weighted composite score and thus the true score variance in a unit-weighted composite based on the respective indicators. Omega total is the total true score variance in a composite. Omega hierarchical is the true score variance in a composite that is attributable to the general factor, and omega subscale is the true score variance in a composite attributable to all subscales / group factors (for the whole scale) or to the specific subscale / group factor (for subscale composites).

Interpretation of omega_subscale

no universal guideline

0.50 is minimum, 0.75 preferable (watkins books,rodrigues16, zimbarg2005

## Omega total, omega hierarchical, omega subscale, H index, explained common variance (ECV), and percent of uncontaminated correlations (PUC) for the general factor (top row) and omegas and H index for the group factors:
## 
##       tot  hier   sub     H   ECV   PUC
## g   0.947 0.565 0.203 0.883 0.249 0.933
## F1  0.820 0.344 0.476 0.773            
## F2  0.739 0.303 0.436 0.586            
## F3  0.816 0.350 0.466 0.724            
## F4  0.798 0.189 0.609 0.802            
## F5  0.519 0.151 0.368 0.368            
## F6  0.772 0.055 0.717 0.868            
## F7  0.665 0.141 0.525 0.624            
## F8  0.701 0.178 0.523 0.705            
## F9  0.827 0.237 0.590 0.692            
## F10 0.577 0.089 0.488 0.573            
## F11 0.830 0.216 0.614 0.731            
## F12 0.822 0.014 0.809 0.831            
## F13 0.798 0.155 0.643 0.791

2.3 14 Factors

2.3.1 Imputation of Missing Items (14 factors)

include reference on why imputation is better than removing observations, pairwise removing is biased, using means isn’t good.

Categorical missing values using multiple Correspondence Analysis (also called Missing Fuzzy Average method) Josseet al (201 0)

2.3.2 14-Factors: Pre-Imputation Tests for Data Suitability for Factor Analysis

2.3.2.1 Bartlett’s Test of Sphericity

This function tests whether a correlation matrix is significantly different from an identity matrix (Bartlett, 1951). If the Bartlett’s test is not significant, the correlation matrix is not suitable for factor analysis because the variables show too little covariance.

## Warning in polychoric(ddNum14, smooth = TRUE, correct = 0.01): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 991 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.5499905^{4} with 1128 DF, p = 0e+00.

2.3.2.2 Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8491539

2.3.3 14-Factor: Post-Imputation Tests for Data Suitability for Factor Analysis

2.3.3.1 Bartlett’s Test of Sphericity

## Warning in polychoric(imputedNumericDF14, smooth = TRUE, correct = 0.01): The
## items do not have an equal number of response alternatives, global set to
## FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 991 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.4280193^{4} with 1128 DF, p = 0e+00.

2.3.3.2 14-Factor: Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8491539

2.3.4 Between-Item Correlations

2.3.4.1 Pre-Imputation

For factor analysis, it is recommended that some of the item correlationsshould be between 0.3 and 0.9. Polychoric correlations were computed using the polychoric function in the psych package in R with options smooth = TRUE to adjust for non-positive-definite matrices and correct = 0.01 to avoid computation problems with covariance of zero.

The minimum correlation in this data set is -0.477. The maximum correlation in this data set is 0.856.

#### Post-Imputation

The post imputation polyserial correlation was also computed using the polychoric function, with no setting smooth = TRUE and correct = 0.01.

The minimum correlation was -0.466. The maximum correlation was 0.851.

2.3.5 Estimating Number of Factors

2.3.5.1 Parallel method

From the EFAtools documentation:

Various methods for performing parallel analysis. This function uses future_lapply for which a parallel processing plan can be selected. To do so, call library(future) and, for example, plan(multisession); see examples.

Settings Used:
- n.obs = 1117 - eigen_type = “EFA” - fa = “fa” (factor analysis not PCA) - fm = “wls” (weighted least squares because pa gave errors and weighted recommended for ordinal data. - use = “all.obs” (since we have imputed missing values, we can use all data points) - cor = “poly” (use polychoric correlation matrix) - n.iter = 100 (run for 100 iterations)

The suggested number of factors is 11.

2.3.5.2 Minimum Average Partial

MAP is recommended as a way to find the number of factors when the items are ordinal. The lowest value indicates the best number of factors.

Number of Factors MAP value
1 0.0327
2 0.0280
3 0.0219
4 0.0192
5 0.0174
6 0.0168
7 0.0164
8 0.0155
9 0.0143
10 0.0145
11 0.0145
12 0.0145
13 0.0148
14 0.0152
15 0.0163
16 0.0173
17 0.0184
18 0.0197

2.3.6 Factor Analysis for 14 Factors

2.3.6.1 14 Factors Model Fit

## Warning in polychoric(r, correct = correct, weight = weight): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 991 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

Although the chi-square test of goodness of fit is sensitive to departures from normality like the C-BARQ items, Hopper et al (2008) recommend always reporting it.

  • chi-square: 4373.564513
  • degrees of freedom: 547
  • P-value for chi-square = 0e+00

Tucker-Lewis Index of Factoring Reliability/Non-Norm Fit Index: 0.7598549. Should be > 0.9; need reference)

2.3.6.2 14 Factor Model Communalities

fanal14.communality
TRAIN01 0.50
TRAIN03 0.45
TRAIN04 0.59
TRAIN07 0.62
AGG09 0.78
AGG10 0.70
AGG14 0.51
AGG15 0.77
AGG16 0.76
AGG18 0.72
AGG19 0.76
AGG20 0.56
FEAR21 0.83
FEAR23 0.82
FEAR24 0.88
FEAR25 0.44
FEAR26 0.44
FEAR27 0.80
FEAR28 0.65
FEAR29 0.61
FEAR30 0.62
FEAR31 0.72
SEPR32 0.68
SEPR33 0.66
SEPR34 0.65
EXCITE35 0.46
EXCITE36 0.58
EXCITE37 0.64
EXCITE38 0.65
PLAY43 0.55
PLAY44 0.83
PLAY45 0.76
PLAY46 0.74
IMP47 0.59
IMP48 0.60
MISC52 0.62
MISC53 0.55
MISC54 0.65
MISC55 0.71
MISC56 0.62
MISC59 0.57
MISC60 0.58
MISC61 0.71
MISC62 0.45
MISC64 0.60
MISC65 0.65
MISC66 0.69
MISC67 0.60

2.3.6.3 How many communalities < 0.40?

There are 0 items with communality < 0.40.

fanal14.communality item

2.3.6.4 14 Factor Model Loadings

PA1 PA3 PA13 PA2 PA5 PA8 PA7 PA4 PA9 PA10 PA11 PA14 PA12 PA6
TRAIN01 -0.06 0.10 0.00 -0.02 -0.05 -0.06 0.02 0.00 -0.01 0.06 0.61 0.00 0.01 0.11
TRAIN03 -0.01 -0.05 -0.04 -0.04 0.07 0.07 0.10 0.02 -0.05 0.06 0.60 0.05 0.03 0.04
TRAIN04 0.06 0.24 -0.04 -0.03 -0.01 0.03 0.03 -0.06 0.00 -0.02 0.56 0.18 -0.08 -0.03
TRAIN07 0.06 0.62 -0.09 -0.05 -0.06 0.01 0.03 0.00 0.00 0.01 0.27 0.01 -0.05 -0.01
AGG09 0.21 0.02 0.76 0.01 0.07 -0.04 -0.06 -0.02 -0.01 0.04 0.08 0.05 0.07 -0.08
AGG10 -0.07 -0.01 0.83 0.06 -0.03 0.02 0.11 0.01 0.02 0.05 -0.06 0.03 -0.06 -0.01
AGG14 0.04 0.12 0.65 0.07 -0.02 0.07 0.13 0.03 0.00 -0.06 -0.08 0.00 -0.13 0.12
AGG15 0.29 -0.02 0.69 -0.06 0.03 -0.02 -0.06 0.11 -0.13 0.07 0.02 -0.01 0.07 0.05
AGG16 -0.10 0.04 0.31 -0.05 0.07 0.00 -0.04 0.49 0.37 -0.07 0.10 -0.04 0.02 -0.10
AGG18 -0.13 -0.01 0.30 -0.03 0.00 -0.07 -0.03 0.51 0.34 -0.04 0.07 0.04 0.10 -0.06
AGG19 0.06 0.02 0.00 0.07 0.02 0.07 0.02 0.84 -0.01 0.05 0.03 -0.07 0.03 -0.02
AGG20 0.03 -0.01 -0.08 -0.04 0.01 -0.01 0.02 0.72 -0.09 0.07 -0.08 0.18 -0.14 0.09
FEAR21 0.84 0.00 0.08 -0.01 0.02 0.00 -0.01 0.01 0.05 0.07 0.00 0.02 0.02 -0.02
FEAR23 0.80 0.00 0.12 -0.01 0.05 -0.02 0.07 0.00 0.12 -0.06 0.00 0.00 0.01 -0.04
FEAR24 0.87 0.01 0.08 -0.07 -0.03 0.04 -0.02 0.01 0.11 0.01 -0.02 -0.04 0.01 0.02
FEAR25 0.36 0.20 -0.11 0.08 0.29 0.03 0.06 0.08 0.05 -0.03 0.08 0.04 -0.10 0.05
FEAR26 0.40 0.00 -0.01 0.07 -0.02 -0.03 0.04 0.11 0.05 0.39 0.01 0.00 0.01 0.06
FEAR27 0.20 0.00 -0.04 0.01 0.02 -0.01 0.02 0.06 0.79 0.02 -0.02 0.01 -0.03 0.05
FEAR28 0.48 0.13 -0.10 0.06 0.36 0.05 0.04 0.01 0.04 0.14 0.01 0.04 -0.08 -0.10
FEAR29 -0.07 -0.02 0.08 -0.04 -0.04 0.02 0.00 -0.04 0.05 0.80 0.04 -0.02 0.03 0.00
FEAR30 -0.01 0.04 -0.02 0.01 0.08 -0.01 0.07 0.10 0.02 0.70 -0.02 0.03 -0.04 0.04
FEAR31 0.09 0.01 -0.07 -0.02 0.05 0.02 0.03 -0.07 0.78 0.12 -0.01 0.06 -0.06 0.03
SEPR32 0.08 0.01 0.02 -0.06 -0.05 0.08 0.78 0.05 0.00 0.04 -0.07 0.03 0.01 -0.08
SEPR33 -0.06 0.03 0.00 0.02 0.02 0.00 0.80 -0.03 0.04 0.03 0.03 -0.01 0.01 0.01
SEPR34 -0.03 -0.01 0.05 0.01 0.06 -0.04 0.78 -0.03 0.00 -0.01 0.08 0.00 0.04 0.06
EXCITE35 0.13 0.04 -0.03 -0.09 -0.09 0.59 0.14 0.03 0.00 -0.04 0.01 -0.05 0.14 0.05
EXCITE36 0.01 0.03 0.01 0.13 0.02 0.65 -0.06 -0.01 -0.07 0.06 -0.03 0.03 0.13 -0.04
EXCITE37 -0.03 0.01 0.04 -0.06 -0.02 0.81 0.02 0.01 0.02 -0.01 0.01 0.01 0.00 0.01
EXCITE38 -0.03 -0.09 -0.01 0.05 0.06 0.77 -0.02 0.02 0.02 0.00 0.02 0.07 -0.05 0.01
PLAY43 -0.28 0.02 -0.24 0.50 0.01 0.07 -0.04 -0.07 0.09 -0.01 -0.08 0.02 0.07 0.03
PLAY44 0.00 0.03 -0.03 0.85 -0.03 0.04 -0.02 0.09 0.00 0.04 0.00 0.06 0.03 -0.10
PLAY45 0.02 0.01 0.04 0.82 -0.04 -0.02 -0.05 -0.06 0.06 -0.02 -0.09 0.01 0.09 0.02
PLAY46 0.01 -0.13 0.10 0.82 0.02 0.01 0.06 0.00 -0.09 -0.05 0.07 -0.01 -0.02 0.01
IMP47 0.01 -0.07 0.04 0.11 -0.02 0.07 -0.02 0.00 0.05 -0.01 0.16 0.64 0.06 0.07
IMP48 -0.09 0.03 0.08 0.00 -0.01 0.09 0.03 0.00 0.06 0.02 0.00 0.71 0.00 0.03
MISC52 0.00 0.06 0.07 -0.02 0.04 0.04 -0.04 -0.03 0.08 0.01 -0.04 -0.02 0.00 0.76
MISC53 -0.01 -0.06 -0.06 -0.03 -0.01 -0.02 0.04 0.07 -0.03 0.02 0.11 0.08 0.02 0.68
MISC54 0.00 0.16 0.00 -0.03 0.69 0.07 -0.02 -0.01 0.01 0.08 -0.01 0.00 -0.09 0.06
MISC55 0.00 -0.06 0.00 0.03 0.86 0.00 0.06 0.04 0.00 -0.01 0.01 -0.07 0.01 -0.02
MISC56 -0.06 -0.01 0.04 -0.09 0.78 -0.04 -0.04 -0.03 0.02 0.01 -0.01 0.08 0.08 0.03
MISC59 0.07 0.09 -0.09 0.03 0.00 -0.06 0.26 0.10 -0.09 0.01 -0.02 0.41 0.34 0.04
MISC60 -0.02 0.01 -0.06 0.04 0.06 0.08 0.00 -0.12 0.02 0.04 0.03 0.05 0.70 0.03
MISC61 0.05 0.00 0.00 0.15 -0.04 0.10 0.11 0.04 -0.10 -0.05 -0.05 0.03 0.69 -0.03
MISC62 -0.09 0.38 0.05 -0.06 0.00 0.03 -0.02 0.16 0.32 -0.01 0.06 0.01 0.22 0.04
MISC64 -0.10 0.78 0.06 0.00 0.00 -0.03 0.03 0.00 -0.03 0.02 0.01 -0.07 0.02 0.04
MISC65 -0.09 0.71 0.03 -0.02 -0.02 -0.01 0.02 0.06 0.05 0.03 0.10 0.01 0.02 0.07
MISC66 0.19 0.62 0.00 -0.05 0.17 -0.04 0.03 -0.07 -0.02 0.04 0.03 0.11 -0.04 -0.03
MISC67 0.16 0.60 0.02 -0.13 0.14 -0.02 0.04 -0.01 0.02 0.02 -0.10 0.10 0.01 -0.02
2.3.6.4.1 14 Factor Model Largest Loading Per Item
Largest Loading Per Item and Associated Factors, Sorted by Factor
largest maximum itemNames itemDescriptions
13 PA1 0.84 FEAR21 When an unfamiliar person approaches the dog when s/he is away from his/her normal home environment or kennel
14 PA1 0.80 FEAR23 When an unfamiliar person visits your home or approaches the dog when in his/her home kennel
15 PA1 0.87 FEAR24 When an unfamiliar person tries to touch or pet the dog
16 PA1 0.36 FEAR25 In response to strange or unfamiliar objects on or near sidewalks or walkways (e.g., plastic trash bags, leaves, litter, flags flapping, etc.
17 PA1 0.40 FEAR26 When examined/treated by a veterinarian
19 PA1 0.48 FEAR28 When first exposed to unfamiliar situations (e.g., novel environments, first visit to the veterinarian, etc.)
20 PA10 0.80 FEAR29 When having nails trimmed, or feet touched/handled
21 PA10 0.70 FEAR30 When groomed or bathed
45 PA11 0.61 TRAIN01 Is hard to recall when off the leash
46 PA11 0.60 TRAIN03 Is slow to obey a ‘stay’ command
47 PA11 0.56 TRAIN04 Has difficulty attending/listening to things you say or do
31 PA12 0.70 MISC60 Playful, puppyish, boisterous
32 PA12 0.69 MISC61 Active, energetic, always on the go
1 PA13 0.76 AGG09 Barks, growls, attempts to bite when approached directly by an unfamiliar person while being walked/exercised on a leash
2 PA13 0.83 AGG10 Barks, etc. when unfamiliar persons approach the dog when s/he is in his/her kennel
3 PA13 0.65 AGG14 When strangers walk past when the dog is in his/her home run or kennel
4 PA13 0.69 AGG15 When an unfamiliar person tries to touch or pet the dog
23 PA14 0.64 IMP47 Impulsive; doesn’t seem to think before s/he acts
24 PA14 0.71 IMP48 Becomes frustrated/impatient in a wide range of situations
30 PA14 0.41 MISC59 Hyperactive, restless, has trouble settling down
38 PA2 0.50 PLAY43 Eagerly engages in play with new/unfamiliar people
39 PA2 0.85 PLAY44 Highly toy focused; attention riveted on tug toy/balls when these are held by handler or other person
40 PA2 0.82 PLAY45 Eagerly initiates play sessions; brings objects/toys to you/the handler and retrieves them when thrown
41 PA2 0.82 PLAY46 Hunts persistently for thrown or hidden toys/objects, not easily distracted from this task
33 PA3 0.38 MISC62 Becomes highly excited and/or distracted when encountering unfamiliar dogs
34 PA3 0.78 MISC64 When working, is easily distracted or preoccupied by odors/engages in persistent sniffing of ground or objects
35 PA3 0.71 MISC65 Has difficulty shifting attention away from interesting or distracting stimuli (e.g., other dogs, odor, people, small animals, etc.)
36 PA3 0.62 MISC66 Is distracted or nervous in new, unfamiliar environments, has difficulty maintaining focus on work
37 PA3 0.60 MISC67 Is slow to recover after being distracted, startled, or frightened/takes a long time to resume work
48 PA3 0.62 TRAIN07 Unfocused; is easily distracted by interesting sights, sounds or smells
5 PA4 0.49 AGG16 When approached directly by an unfamiliar dog while being walked/exercised on a leash
6 PA4 0.51 AGG18 When barked, growled, or lunged at by another dog
7 PA4 0.84 AGG19 Towards other familiar dogs
8 PA4 0.72 AGG20 When approached while playing with/chewing a favorite toy, bone, object, etc. by another familiar dog
27 PA5 0.69 MISC54 Reluctant to/nervous about crossing grates or other unfamiliar surfaces
28 PA5 0.86 MISC55 Reluctant to/nervous about crossing shiny or slippery floors
29 PA5 0.78 MISC56 Nervous or frightened when ascending or descending some types of stairs
25 PA6 0.76 MISC52 Begs persistently for food when people are eating
26 PA6 0.68 MISC53 Steals food
42 PA7 0.78 SEPR32 Restlessness/agitation/pacing when left alone
43 PA7 0.80 SEPR33 Barking or whining when left alone
44 PA7 0.78 SEPR34 Chewing/scratching at doors, floor, fencing, etc., when left alone
9 PA8 0.59 EXCITE35 Excitable when you first arrive home, or at the dog’s kennel, after a brief absence
10 PA8 0.65 EXCITE36 Excitable when playing with you or other familiar persons
11 PA8 0.81 EXCITE37 Excitable just before being taken out for a walk
12 PA8 0.77 EXCITE38 Excitable just before being taken out for work or training
18 PA9 0.79 FEAR27 When approached directly by an unfamiliar dog
22 PA9 0.78 FEAR31 When barked, growled, or lunged at by an unfamiliar dog

2.3.7 14 Model Reliability Measures

PA1 PA3 PA13 PA2 PA5 PA8 PA7 PA4 PA9 PA10 PA11 PA14 PA12 PA6
alpha 0.797 0.787 0.76 0.812 0.718 0.787 0.735 0.753 0.802 0.616 0.652 0.682 0.71 0.623

omega is thought to be better measure of reliability than alpha, because it looks at the relationships between all the items rather than just the ones within each factor. It’s different than alpha because it assumes that there is a genral factor, g, that all items have in common. This is listed as the first row of the table below.

There are three omegas for each factor:

  • omega_total - total true score variance in factor
  • omega_hierarchical - variance attributable to general factor (g)
  • omega_subscale - variance attributable to the subscale

From EFAtools::OMEGA documentation:

Omegas refer to the correlation between a factor and a unit-weighted composite score and thus the true score variance in a unit-weighted composite based on the respective indicators. Omega total is the total true score variance in a composite. Omega hierarchical is the true score variance in a composite that is attributable to the general factor, and omega subscale is the true score variance in a composite attributable to all subscales / group factors (for the whole scale) or to the specific subscale / group factor (for subscale composites).

Interpretation of omega_subscale

no universal guideline

0.50 is minimum, 0.75 preferable (watkins books,rodrigues16, zimbarg2005

## Omega total, omega hierarchical, omega subscale, H index, explained common variance (ECV), and percent of uncontaminated correlations (PUC) for the general factor (top row) and omegas and H index for the group factors:
## 
##       tot  hier   sub     H   ECV   PUC
## g   0.946 0.566 0.208 0.879 0.229 0.938
## F1  0.834 0.326 0.508 0.818            
## F2  0.752 0.230 0.522 0.596            
## F3  0.641 0.164 0.477 0.573            
## F4  0.596 0.001 0.595 0.649            
## F5  0.814 0.120 0.694 0.816            
## F6  0.597 0.159 0.438 0.588            
## F7  0.809 0.065 0.744 0.858            
## F8  0.829 0.387 0.442 0.702            
## F9  0.703 0.157 0.546 0.759            
## F10 0.835 0.199 0.637 0.749            
## F11 0.709 0.078 0.630 0.662            
## F12 0.843 0.162 0.681 0.764            
## F13 0.774 0.021 0.754 0.818            
## F14 0.802 0.214 0.588 0.688

2.4 15 Factors

2.4.1 Imputation of Missing Items (15 factors)

include reference on why imputation is better than removing observations, pairwise removing is biased, using means isn’t good.

Categorical missing values using multiple Correspondence Analysis (also called Missing Fuzzy Average method) Josseet al (201 0)

2.4.2 15-Factors: Pre-Imputation Tests for Data Suitability for Factor Analysis

2.4.2.1 Bartlett’s Test of Sphericity

This function tests whether a correlation matrix is significantly different from an identity matrix (Bartlett, 1951). If the Bartlett’s test is not significant, the correlation matrix is not suitable for factor analysis because the variables show too little covariance.

## Warning in polychoric(ddNum15, smooth = TRUE, correct = 0.01): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1102 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.6576356^{4} with 1275 DF, p = 0e+00.

2.4.2.2 Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8460796

2.4.3 15-Factor: Post-Imputation Tests for Data Suitability for Factor Analysis

2.4.3.1 Bartlett’s Test of Sphericity

## Warning in polychoric(imputedNumericDF15, smooth = TRUE, correct = 0.01): The
## items do not have an equal number of response alternatives, global set to
## FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1102 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.5265375^{4} with 1275 DF, p = 0e+00.

2.4.3.2 15-Factor: Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8460796

2.4.4 Between-Item Correlations

2.4.4.1 Pre-Imputation

For factor analysis, it is recommended that some of the item correlationsshould be between 0.3 and 0.9. Polychoric correlations were computed using the polychoric function in the psych package in R with options smooth = TRUE to adjust for non-positive-definite matrices and correct = 0.01 to avoid computation problems with covariance of zero.

The minimum correlation in this data set is -0.477. The maximum correlation in this data set is 0.856.

#### Post-Imputation

The post imputation polyserial correlation was also computed using the polychoric function, with no setting smooth = TRUE and correct = 0.01.

The minimum correlation was -0.466. The maximum correlation was 0.851.

2.4.5 Estimating Number of Factors

2.4.5.1 Parallel method

From the EFAtools documentation:

Various methods for performing parallel analysis. This function uses future_lapply for which a parallel processing plan can be selected. To do so, call library(future) and, for example, plan(multisession); see examples.

Settings Used:
- n.obs = 1117 - eigen_type = “EFA” - fa = “fa” (factor analysis not PCA) - fm = “wls” (weighted least squares because pa gave errors and weighted recommended for ordinal data. - use = “all.obs” (since we have imputed missing values, we can use all data points) - cor = “poly” (use polychoric correlation matrix) - n.iter = 100 (run for 100 iterations)

The suggested number of factors is 12.

2.4.5.2 Minimum Average Partial

MAP is recommended as a way to find the number of factors when the items are ordinal. The lowest value indicates the best number of factors.

Number of Factors MAP value
1 0.0302
2 0.0256
3 0.0198
4 0.0176
5 0.0163
6 0.0158
7 0.0157
8 0.0147
9 0.0142
10 0.0137
11 0.0134
12 0.0137
13 0.0138
14 0.0141
15 0.0150
16 0.0154
17 0.0163
18 0.0176

2.4.6 Factor Analysis for 15 Factors

2.4.6.1 15 Factors Model Fit

## Warning in polychoric(r, correct = correct, weight = weight): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1102 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

Although the chi-square test of goodness of fit is sensitive to departures from normality like the C-BARQ items, Hopper et al (2008) recommend always reporting it.

  • chi-square: 4709.9196124
  • degrees of freedom: 615
  • P-value for chi-square = 0e+00

Tucker-Lewis Index of Factoring Reliability/Non-Norm Fit Index: 0.747881. Should be > 0.9; need reference)

2.4.6.2 15 Factor Model Communalities

fanal15.communality
TRAIN01 0.44
TRAIN02 0.41
TRAIN03 0.60
TRAIN04 0.61
TRAIN07 0.62
AGG09 0.76
AGG10 0.70
AGG14 0.51
AGG15 0.80
AGG16 0.76
AGG18 0.74
AGG19 0.77
AGG20 0.52
FEAR21 0.84
FEAR23 0.81
FEAR24 0.89
FEAR26 0.46
FEAR27 0.85
FEAR28 0.63
FEAR29 0.53
FEAR30 0.62
FEAR31 0.71
SEPR32 0.68
SEPR33 0.65
SEPR34 0.67
EXCITE35 0.48
EXCITE36 0.58
EXCITE37 0.65
EXCITE38 0.66
ATT39 0.52
ATT41 0.33
PLAY43 0.55
PLAY44 0.84
PLAY45 0.76
PLAY46 0.72
IMP47 0.57
IMP48 0.62
MISC52 0.52
MISC53 0.61
MISC54 0.64
MISC55 0.79
MISC56 0.60
MISC59 0.59
MISC60 0.61
MISC61 0.70
MISC62 0.45
MISC64 0.61
MISC65 0.65
MISC66 0.69
MISC67 0.59
MISC69 0.20

2.4.6.3 How many communalities < 0.40?

There are 2 items with communality < 0.40.

fanal15.communality item
ATT41 0.33 ATT41
MISC69 0.20 MISC69

2.4.6.4 15 Factor Model Loadings

PA1 PA3 PA12 PA2 PA5 PA6 PA7 PA4 PA9 PA8 PA11 PA13 PA15 PA10 PA14
TRAIN01 -0.09 0.22 0.03 -0.05 -0.09 -0.05 0.01 0.01 0.01 0.46 0.04 0.04 0.01 0.12 -0.14
TRAIN02 0.01 0.01 0.02 0.03 0.05 0.02 0.01 -0.09 0.02 0.61 0.07 -0.06 0.05 -0.02 0.06
TRAIN03 0.00 -0.06 -0.04 -0.03 0.05 0.01 0.06 0.04 -0.07 0.75 0.05 0.04 0.02 0.03 0.04
TRAIN04 0.08 0.33 -0.03 -0.01 -0.03 0.04 0.03 -0.04 0.02 0.48 -0.07 -0.04 0.13 0.02 -0.21
TRAIN07 0.08 0.67 -0.10 -0.04 -0.05 0.00 0.02 0.02 0.00 0.21 -0.02 -0.05 0.00 0.00 -0.01
AGG09 0.21 0.04 0.74 0.01 0.05 -0.05 -0.05 0.01 -0.01 0.04 0.01 0.07 0.05 -0.07 0.02
AGG10 -0.07 -0.04 0.83 0.06 -0.03 0.01 0.11 0.01 0.02 -0.01 0.05 -0.07 0.04 -0.03 0.05
AGG14 0.04 0.10 0.65 0.07 0.00 0.06 0.13 0.03 0.01 -0.06 -0.08 -0.14 0.00 0.12 0.06
AGG15 0.28 0.00 0.73 -0.07 0.02 0.01 -0.06 0.10 -0.09 -0.01 0.08 0.09 -0.03 0.04 -0.12
AGG16 -0.11 0.04 0.29 -0.06 0.06 -0.03 -0.06 0.51 0.33 0.10 -0.08 0.01 0.01 -0.13 0.09
AGG18 -0.14 -0.01 0.28 -0.03 0.00 -0.11 -0.05 0.54 0.29 0.07 -0.05 0.08 0.08 -0.08 0.14
AGG19 0.06 0.03 0.00 0.07 0.02 0.08 0.03 0.86 -0.02 0.00 0.06 0.01 -0.08 0.00 -0.05
AGG20 0.04 -0.02 -0.08 -0.03 0.02 0.00 0.03 0.69 -0.09 -0.06 0.10 -0.15 0.18 0.10 0.00
FEAR21 0.84 0.01 0.07 0.01 0.04 -0.01 0.00 0.02 0.06 0.00 0.06 0.01 0.02 0.00 0.02
FEAR23 0.77 0.00 0.12 -0.01 0.06 -0.03 0.06 -0.01 0.13 0.00 -0.05 0.00 0.02 -0.05 0.03
FEAR24 0.86 0.01 0.08 -0.06 -0.02 0.03 -0.02 0.01 0.11 0.00 0.02 -0.01 -0.04 0.01 0.04
FEAR26 0.36 0.00 0.02 0.05 -0.03 -0.02 0.04 0.08 0.07 0.04 0.43 0.01 0.01 0.03 0.02
FEAR27 0.19 0.00 -0.04 0.02 0.02 0.01 0.03 0.05 0.85 -0.03 0.00 -0.02 -0.01 0.05 -0.04
FEAR28 0.43 0.11 -0.08 0.02 0.34 0.05 0.03 -0.02 0.05 0.02 0.18 -0.06 0.10 -0.15 0.03
FEAR29 -0.06 0.00 0.08 -0.02 -0.03 0.01 0.03 -0.01 0.06 0.07 0.71 0.04 -0.04 0.04 0.01
FEAR30 -0.01 0.04 -0.01 0.02 0.08 -0.01 0.09 0.10 0.04 0.00 0.68 -0.03 0.03 0.06 -0.04
FEAR31 0.07 0.02 -0.07 -0.02 0.04 0.02 0.03 -0.07 0.78 -0.01 0.12 -0.04 0.06 0.00 0.01
SEPR32 0.07 -0.01 0.01 -0.06 -0.05 0.07 0.77 0.05 -0.01 -0.03 0.05 -0.01 0.06 -0.11 0.07
SEPR33 -0.07 0.03 0.00 0.01 0.02 -0.01 0.78 -0.03 0.04 0.07 0.04 0.00 0.00 0.00 0.05
SEPR34 -0.03 0.03 0.07 0.00 0.06 -0.03 0.79 -0.03 0.03 0.03 -0.01 0.06 -0.02 0.09 -0.08
EXCITE35 0.14 0.02 -0.05 -0.08 -0.07 0.55 0.13 0.04 -0.01 0.08 -0.06 0.11 -0.04 0.03 0.16
EXCITE36 -0.01 0.00 0.01 0.12 0.01 0.64 -0.06 -0.01 -0.07 0.01 0.08 0.13 0.04 -0.08 0.07
EXCITE37 -0.01 0.04 0.04 -0.05 -0.02 0.82 0.02 0.03 0.03 -0.01 -0.03 0.00 0.00 0.02 -0.02
EXCITE38 -0.04 -0.08 0.00 0.04 0.05 0.79 -0.02 0.01 0.04 -0.01 0.00 -0.02 0.06 0.02 -0.04
ATT39 0.11 0.03 0.01 0.04 -0.04 0.06 0.03 -0.03 -0.07 -0.01 -0.01 0.07 -0.08 0.05 0.67
ATT41 0.06 0.06 -0.14 0.05 0.02 -0.02 0.09 0.07 -0.05 0.03 -0.05 0.04 0.15 0.19 0.44
PLAY43 -0.28 0.01 -0.23 0.50 0.02 0.08 -0.04 -0.08 0.11 -0.06 -0.01 0.07 0.00 0.04 -0.02
PLAY44 0.02 0.04 -0.04 0.88 -0.02 0.03 -0.02 0.11 -0.01 0.01 0.03 0.02 0.03 -0.07 0.00
PLAY45 0.01 -0.02 0.04 0.83 -0.03 -0.03 -0.06 -0.07 0.06 -0.05 0.01 0.07 0.03 -0.01 0.08
PLAY46 0.00 -0.10 0.11 0.80 0.02 0.02 0.05 -0.01 -0.07 0.02 -0.05 -0.01 0.00 0.01 -0.05
IMP47 0.04 -0.04 0.03 0.15 -0.02 0.07 -0.01 0.01 0.04 0.16 -0.04 0.08 0.58 0.14 -0.05
IMP48 -0.08 -0.01 0.07 0.02 -0.02 0.10 0.04 0.00 0.03 0.04 0.03 0.00 0.72 0.03 0.00
MISC52 -0.06 0.05 0.11 -0.10 0.04 0.05 -0.08 -0.09 0.10 -0.02 0.10 -0.02 0.05 0.61 0.14
MISC53 0.02 -0.01 -0.03 -0.03 0.01 -0.01 0.02 0.06 0.00 0.07 0.03 0.00 0.05 0.74 -0.01
MISC54 -0.02 0.16 -0.01 -0.04 0.67 0.06 -0.04 -0.03 0.01 0.01 0.11 -0.10 0.05 0.02 0.07
MISC55 0.02 -0.05 -0.02 0.05 0.91 -0.01 0.06 0.05 0.00 0.04 -0.05 0.01 -0.08 0.00 -0.01
MISC56 -0.05 0.01 0.05 -0.10 0.74 -0.01 -0.04 -0.03 0.03 -0.02 0.03 0.08 0.07 0.02 -0.05
MISC59 0.08 0.10 -0.09 0.04 -0.01 -0.04 0.28 0.10 -0.08 -0.07 0.01 0.36 0.39 0.08 -0.08
MISC60 -0.04 -0.01 -0.05 0.02 0.04 0.06 -0.01 -0.11 0.02 0.04 0.04 0.72 0.04 0.01 0.07
MISC61 0.04 -0.02 0.02 0.14 -0.04 0.10 0.11 0.03 -0.09 -0.04 -0.04 0.68 0.01 -0.03 0.02
MISC62 -0.11 0.39 0.06 -0.08 -0.01 0.03 -0.04 0.17 0.30 0.03 0.00 0.22 0.03 0.00 0.03
MISC64 -0.08 0.80 0.05 0.02 0.01 -0.02 0.04 0.00 -0.03 -0.02 0.01 0.01 -0.09 0.04 0.02
MISC65 -0.09 0.75 0.03 -0.02 -0.01 0.00 0.02 0.06 0.06 0.03 0.03 0.02 0.00 0.07 -0.01
MISC66 0.18 0.62 0.00 -0.07 0.17 -0.04 0.03 -0.08 -0.02 0.00 0.07 -0.04 0.14 -0.06 0.02
MISC67 0.15 0.57 0.01 -0.13 0.13 -0.02 0.02 -0.03 0.00 -0.07 0.06 -0.01 0.14 -0.08 0.08
MISC69 0.10 0.12 -0.01 0.02 0.24 0.02 0.13 0.08 0.10 0.00 -0.24 0.05 0.01 0.18 -0.02
2.4.6.4.1 15 Factor Model Largest Loading Per Item
Largest Loading Per Item and Associated Factors, Sorted by Factor
largest maximum itemNames itemDescriptions
15 PA1 0.84 FEAR21 When an unfamiliar person approaches the dog when s/he is away from his/her normal home environment or kennel
16 PA1 0.77 FEAR23 When an unfamiliar person visits your home or approaches the dog when in his/her home kennel
17 PA1 0.86 FEAR24 When an unfamiliar person tries to touch or pet the dog
20 PA1 0.43 FEAR28 When first exposed to unfamiliar situations (e.g., novel environments, first visit to the veterinarian, etc.)
26 PA10 0.61 MISC52 Begs persistently for food when people are eating
27 PA10 0.74 MISC53 Steals food
18 PA11 0.43 FEAR26 When examined/treated by a veterinarian
21 PA11 0.71 FEAR29 When having nails trimmed, or feet touched/handled
22 PA11 0.68 FEAR30 When groomed or bathed
1 PA12 0.74 AGG09 Barks, growls, attempts to bite when approached directly by an unfamiliar person while being walked/exercised on a leash
2 PA12 0.83 AGG10 Barks, etc. when unfamiliar persons approach the dog when s/he is in his/her kennel
3 PA12 0.65 AGG14 When strangers walk past when the dog is in his/her home run or kennel
4 PA12 0.73 AGG15 When an unfamiliar person tries to touch or pet the dog
32 PA13 0.72 MISC60 Playful, puppyish, boisterous
33 PA13 0.68 MISC61 Active, energetic, always on the go
9 PA14 0.67 ATT39 Displays a strong attachment for you or another familiar person
10 PA14 0.44 ATT41 Tends to nudge or paw you (or others) for attention
24 PA15 0.58 IMP47 Impulsive; doesn’t seem to think before s/he acts
25 PA15 0.72 IMP48 Becomes frustrated/impatient in a wide range of situations
31 PA15 0.39 MISC59 Hyperactive, restless, has trouble settling down
40 PA2 0.50 PLAY43 Eagerly engages in play with new/unfamiliar people
41 PA2 0.88 PLAY44 Highly toy focused; attention riveted on tug toy/balls when these are held by handler or other person
42 PA2 0.83 PLAY45 Eagerly initiates play sessions; brings objects/toys to you/the handler and retrieves them when thrown
43 PA2 0.80 PLAY46 Hunts persistently for thrown or hidden toys/objects, not easily distracted from this task
34 PA3 0.39 MISC62 Becomes highly excited and/or distracted when encountering unfamiliar dogs
35 PA3 0.80 MISC64 When working, is easily distracted or preoccupied by odors/engages in persistent sniffing of ground or objects
36 PA3 0.75 MISC65 Has difficulty shifting attention away from interesting or distracting stimuli (e.g., other dogs, odor, people, small animals, etc.)
37 PA3 0.62 MISC66 Is distracted or nervous in new, unfamiliar environments, has difficulty maintaining focus on work
38 PA3 0.57 MISC67 Is slow to recover after being distracted, startled, or frightened/takes a long time to resume work
51 PA3 0.67 TRAIN07 Unfocused; is easily distracted by interesting sights, sounds or smells
5 PA4 0.51 AGG16 When approached directly by an unfamiliar dog while being walked/exercised on a leash
6 PA4 0.54 AGG18 When barked, growled, or lunged at by another dog
7 PA4 0.86 AGG19 Towards other familiar dogs
8 PA4 0.69 AGG20 When approached while playing with/chewing a favorite toy, bone, object, etc. by another familiar dog
28 PA5 0.67 MISC54 Reluctant to/nervous about crossing grates or other unfamiliar surfaces
29 PA5 0.91 MISC55 Reluctant to/nervous about crossing shiny or slippery floors
30 PA5 0.74 MISC56 Nervous or frightened when ascending or descending some types of stairs
39 PA5 0.24 MISC69 Chases/follows shadows, light spots, etc.
11 PA6 0.55 EXCITE35 Excitable when you first arrive home, or at the dog’s kennel, after a brief absence
12 PA6 0.64 EXCITE36 Excitable when playing with you or other familiar persons
13 PA6 0.82 EXCITE37 Excitable just before being taken out for a walk
14 PA6 0.79 EXCITE38 Excitable just before being taken out for work or training
44 PA7 0.77 SEPR32 Restlessness/agitation/pacing when left alone
45 PA7 0.78 SEPR33 Barking or whining when left alone
46 PA7 0.79 SEPR34 Chewing/scratching at doors, floor, fencing, etc., when left alone
47 PA8 0.46 TRAIN01 Is hard to recall when off the leash
48 PA8 0.61 TRAIN02 Is slow to obey a ‘sit’ command
49 PA8 0.75 TRAIN03 Is slow to obey a ‘stay’ command
50 PA8 0.48 TRAIN04 Has difficulty attending/listening to things you say or do
19 PA9 0.85 FEAR27 When approached directly by an unfamiliar dog
23 PA9 0.78 FEAR31 When barked, growled, or lunged at by an unfamiliar dog

2.4.7 15 Model Reliability Measures

PA1 PA3 PA12 PA2 PA5 PA6 PA7 PA4 PA9 PA8 PA11 PA13 PA15 PA10 PA14
alpha 0.836 0.786 0.76 0.812 0.625 0.787 0.735 0.752 0.803 0.693 0.615 0.71 0.683 0.623 0.439

omega is thought to be better measure of reliability than alpha, because it looks at the relationships between all the items rather than just the ones within each factor. It’s different than alpha because it assumes that there is a genral factor, g, that all items have in common. This is listed as the first row of the table below.

There are three omegas for each factor:

  • omega_total - total true score variance in factor
  • omega_hierarchical - variance attributable to general factor (g)
  • omega_subscale - variance attributable to the subscale

From EFAtools::OMEGA documentation:

Omegas refer to the correlation between a factor and a unit-weighted composite score and thus the true score variance in a unit-weighted composite based on the respective indicators. Omega total is the total true score variance in a composite. Omega hierarchical is the true score variance in a composite that is attributable to the general factor, and omega subscale is the true score variance in a composite attributable to all subscales / group factors (for the whole scale) or to the specific subscale / group factor (for subscale composites).

Interpretation of omega_subscale

no universal guideline

0.50 is minimum, 0.75 preferable (watkins books,rodrigues16, zimbarg2005

## Omega total, omega hierarchical, omega subscale, H index, explained common variance (ECV), and percent of uncontaminated correlations (PUC) for the general factor (top row) and omegas and H index for the group factors:
## 
##       tot  hier   sub     H   ECV   PUC
## g   0.944 0.574 0.199 0.877 0.221 0.945
## F1  0.828 0.272 0.555 0.802            
## F2  0.644 0.070 0.574 0.630            
## F3  0.717 0.264 0.453 0.587            
## F4  0.815 0.130 0.685 0.808            
## F5  0.555 0.032 0.522 0.675            
## F6  0.488 0.028 0.460 0.510            
## F7  0.564 0.141 0.423 0.518            
## F8  0.816 0.053 0.763 0.873            
## F9  0.841 0.366 0.475 0.727            
## F10 0.723 0.164 0.559 0.767            
## F11 0.799 0.209 0.590 0.780            
## F12 0.765 0.027 0.738 0.820            
## F13 0.841 0.167 0.674 0.760            
## F14 0.660 0.170 0.490 0.654            
## F15 0.831 0.224 0.607 0.707

2.5 16 Factors

2.5.1 Imputation of Missing Items (16 factors)

include reference on why imputation is better than removing observations, pairwise removing is biased, using means isn’t good.

Categorical missing values using multiple Correspondence Analysis (also called Missing Fuzzy Average method) Josseet al (201 0)

2.5.2 16-Factors: Pre-Imputation Tests for Data Suitability for Factor Analysis

2.5.2.1 Bartlett’s Test of Sphericity

This function tests whether a correlation matrix is significantly different from an identity matrix (Bartlett, 1951). If the Bartlett’s test is not significant, the correlation matrix is not suitable for factor analysis because the variables show too little covariance.

## Warning in polychoric(ddNum16, smooth = TRUE, correct = 0.01): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1240 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.917561^{4} with 1431 DF, p = 0e+00.

2.5.2.2 Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.850597

2.5.3 16-Factor: Post-Imputation Tests for Data Suitability for Factor Analysis

2.5.3.1 Bartlett’s Test of Sphericity

## Warning in polychoric(imputedNumericDF16, smooth = TRUE, correct = 0.01): The
## items do not have an equal number of response alternatives, global set to
## FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1240 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.7685238^{4} with 1431 DF, p = 0e+00.

2.5.3.2 16-Factor: Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.850597

2.5.4 Between-Item Correlations

2.5.4.1 Pre-Imputation

For factor analysis, it is recommended that some of the item correlationsshould be between 0.3 and 0.9. Polychoric correlations were computed using the polychoric function in the psych package in R with options smooth = TRUE to adjust for non-positive-definite matrices and correct = 0.01 to avoid computation problems with covariance of zero.

The minimum correlation in this data set is -0.477. The maximum correlation in this data set is 0.856.

#### Post-Imputation

The post imputation polyserial correlation was also computed using the polychoric function, with no setting smooth = TRUE and correct = 0.01.

The minimum correlation was -0.466. The maximum correlation was 0.851.

2.5.5 Estimating Number of Factors

2.5.5.1 Parallel method

From the EFAtools documentation:

Various methods for performing parallel analysis. This function uses future_lapply for which a parallel processing plan can be selected. To do so, call library(future) and, for example, plan(multisession); see examples.

Settings Used:
- n.obs = 1117 - eigen_type = “EFA” - fa = “fa” (factor analysis not PCA) - fm = “wls” (weighted least squares because pa gave errors and weighted recommended for ordinal data. - use = “all.obs” (since we have imputed missing values, we can use all data points) - cor = “poly” (use polychoric correlation matrix) - n.iter = 100 (run for 100 iterations)

The suggested number of factors is 12.

2.5.5.2 Minimum Average Partial

MAP is recommended as a way to find the number of factors when the items are ordinal. The lowest value indicates the best number of factors.

Number of Factors MAP value
1 0.0284
2 0.0243
3 0.0190
4 0.0166
5 0.0151
6 0.0146
7 0.0145
8 0.0136
9 0.0133
10 0.0131
11 0.0127
12 0.0127
13 0.0131
14 0.0131
15 0.0137
16 0.0144
17 0.0151
18 0.0157

2.5.6 Factor Analysis for 16 Factors

2.5.6.1 16 Factors Model Fit

## Warning in polychoric(r, correct = correct, weight = weight): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1240 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

Although the chi-square test of goodness of fit is sensitive to departures from normality like the C-BARQ items, Hopper et al (2008) recommend always reporting it.

  • chi-square: 5426.5677305
  • degrees of freedom: 687
  • P-value for chi-square = 0e+00

Tucker-Lewis Index of Factoring Reliability/Non-Norm Fit Index: 0.7249164. Should be > 0.9; need reference)

2.5.6.2 16 Factor Model Communalities

fanal16.communality
TRAIN01 0.51
TRAIN02 0.45
TRAIN03 0.61
TRAIN04 0.60
TRAIN07 0.63
AGG09 0.77
AGG10 0.69
AGG14 0.52
AGG15 0.80
AGG16 0.76
AGG18 0.74
AGG19 0.79
AGG20 0.52
FEAR21 0.84
FEAR22 0.41
FEAR23 0.81
FEAR24 0.87
FEAR25 0.52
FEAR26 0.44
FEAR27 0.86
FEAR28 0.67
FEAR29 0.65
FEAR30 0.61
FEAR31 0.68
SEPR32 0.69
SEPR33 0.66
SEPR34 0.70
EXCITE35 0.48
EXCITE36 0.57
EXCITE37 0.66
EXCITE38 0.66
ATT39 0.41
ATT41 0.37
PLAY43 0.55
PLAY44 0.81
PLAY45 0.78
PLAY46 0.73
IMP47 0.62
IMP48 0.62
MISC51 0.49
MISC52 0.63
MISC53 0.56
MISC54 0.63
MISC55 0.75
MISC56 0.63
MISC59 0.57
MISC60 0.57
MISC61 0.68
MISC62 0.45
MISC64 0.59
MISC65 0.65
MISC66 0.70
MISC67 0.63
MISC69 0.38

2.5.6.3 How many communalities < 0.40?

There are 2 items with communality < 0.40.

fanal16.communality item
ATT41 0.37 ATT41
MISC69 0.38 MISC69

2.5.6.4 16 Factor Model Loadings

PA9 PA1 PA3 PA2 PA5 PA7 PA6 PA4 PA12 PA8 PA15 PA11 PA13 PA10 PA14 PA16
TRAIN01 -0.03 0.20 -0.01 -0.02 -0.10 -0.01 -0.06 0.05 -0.01 0.43 -0.03 0.06 0.04 0.14 -0.21 0.19
TRAIN02 -0.02 0.01 0.05 0.02 0.05 0.02 0.02 -0.09 0.04 0.64 0.05 0.03 -0.04 0.01 0.05 -0.11
TRAIN03 -0.01 -0.06 -0.03 -0.03 0.04 0.06 0.01 0.05 -0.07 0.76 0.02 0.02 0.05 0.05 0.03 -0.03
TRAIN04 0.05 0.31 -0.02 -0.04 -0.04 0.00 0.04 -0.07 0.02 0.45 0.18 -0.01 -0.07 -0.06 -0.17 0.14
TRAIN07 0.05 0.64 -0.08 -0.07 -0.08 0.01 0.00 -0.01 0.00 0.20 0.02 0.04 -0.05 -0.04 0.02 0.10
AGG09 0.24 0.05 0.73 0.01 0.06 -0.06 -0.04 0.02 -0.02 0.02 0.05 0.06 0.07 -0.08 0.01 0.08
AGG10 -0.05 -0.02 0.80 0.08 -0.01 0.13 0.00 0.06 0.01 0.00 0.02 0.02 -0.07 0.01 0.05 -0.07
AGG14 0.05 0.08 0.64 0.06 -0.01 0.11 0.06 0.04 -0.02 -0.06 0.02 -0.02 -0.16 0.08 0.12 0.13
AGG15 0.27 0.01 0.72 -0.05 0.05 -0.04 0.00 0.13 -0.08 0.00 -0.01 0.04 0.07 0.06 -0.14 -0.09
AGG16 -0.12 0.05 0.30 -0.06 0.06 -0.06 -0.02 0.51 0.33 0.10 0.00 -0.06 0.01 -0.12 0.11 0.04
AGG18 -0.13 -0.01 0.27 -0.03 -0.01 -0.04 -0.10 0.55 0.28 0.06 0.07 -0.02 0.09 -0.07 0.14 0.06
AGG19 0.04 0.01 0.01 0.07 0.00 0.03 0.09 0.86 0.00 0.00 -0.07 0.05 0.03 0.00 -0.07 0.00
AGG20 0.01 -0.05 -0.05 -0.05 -0.01 0.02 -0.01 0.67 -0.08 -0.06 0.19 0.09 -0.15 0.08 0.02 -0.02
FEAR21 0.82 -0.01 0.11 -0.03 0.01 -0.01 0.01 -0.01 0.08 -0.02 0.02 0.10 0.03 -0.03 0.01 0.04
FEAR22 0.36 0.16 -0.27 0.01 0.20 0.04 0.01 0.22 -0.09 0.06 -0.03 0.00 -0.14 0.08 0.16 -0.06
FEAR23 0.77 -0.01 0.15 -0.03 0.04 0.06 -0.02 -0.01 0.15 0.00 0.01 -0.04 0.01 -0.05 0.03 0.01
FEAR24 0.81 0.00 0.13 -0.09 -0.04 -0.01 0.04 -0.02 0.16 0.00 -0.03 0.03 0.02 0.00 0.02 -0.06
FEAR25 0.41 0.20 -0.15 0.08 0.26 0.03 0.00 0.12 -0.01 0.06 0.06 0.00 -0.15 0.05 0.14 0.14
FEAR26 0.35 0.00 0.02 0.06 -0.02 0.06 -0.02 0.08 0.09 0.04 0.01 0.37 0.01 0.06 -0.02 -0.14
FEAR27 0.16 0.00 -0.04 0.02 0.03 0.03 0.01 0.04 0.86 -0.02 0.01 0.02 -0.03 0.04 -0.02 0.02
FEAR28 0.49 0.13 -0.10 0.06 0.34 0.05 0.04 0.03 0.04 0.05 0.04 0.12 -0.08 -0.07 0.03 -0.10
FEAR29 -0.05 -0.01 0.05 -0.03 -0.03 -0.01 0.01 -0.03 0.02 0.02 -0.02 0.83 0.03 -0.01 0.02 0.04
FEAR30 0.00 0.03 -0.04 0.02 0.10 0.07 -0.02 0.10 0.01 -0.02 0.05 0.67 -0.05 0.04 -0.01 -0.05
FEAR31 0.10 0.04 -0.11 0.01 0.05 0.03 0.01 -0.02 0.75 0.00 0.03 0.09 -0.05 0.06 0.00 -0.03
SEPR32 0.05 0.00 0.03 -0.06 -0.05 0.79 0.07 0.04 0.01 -0.02 0.04 0.03 0.01 -0.09 0.05 -0.09
SEPR33 -0.08 0.02 0.00 0.02 0.02 0.79 -0.01 -0.03 0.05 0.08 0.00 0.02 -0.01 0.01 0.05 -0.04
SEPR34 -0.02 0.01 0.04 0.01 0.05 0.79 -0.03 -0.03 0.01 0.01 -0.01 0.02 0.03 0.05 -0.08 0.14
EXCITE35 0.11 0.01 -0.02 -0.09 -0.09 0.13 0.57 0.02 0.01 0.08 -0.04 -0.05 0.14 0.04 0.12 -0.01
EXCITE36 0.01 0.02 0.00 0.15 0.03 -0.05 0.64 0.01 -0.07 0.02 0.03 0.04 0.13 -0.03 0.03 -0.07
EXCITE37 -0.04 0.03 0.05 -0.06 -0.02 0.02 0.83 0.00 0.04 -0.02 0.01 -0.01 0.00 0.00 -0.04 0.01
EXCITE38 -0.01 -0.08 -0.03 0.06 0.06 -0.03 0.77 0.03 0.01 0.00 0.07 0.01 -0.07 0.01 0.00 0.05
ATT39 0.06 0.03 0.09 -0.01 -0.09 0.05 0.11 -0.08 -0.03 -0.04 -0.11 0.03 0.17 0.07 0.54 -0.02
ATT41 0.05 0.03 -0.11 0.00 -0.05 0.07 -0.01 0.03 -0.08 0.00 0.14 0.05 0.09 0.16 0.47 0.17
PLAY43 -0.28 0.00 -0.25 0.51 0.03 -0.04 0.07 -0.07 0.08 -0.05 0.02 -0.02 0.05 0.03 0.01 0.01
PLAY44 -0.02 0.02 -0.01 0.84 -0.04 -0.01 0.05 0.08 0.01 0.00 0.05 0.04 0.04 -0.10 -0.02 0.00
PLAY45 0.01 -0.01 0.05 0.84 -0.04 -0.04 -0.03 -0.06 0.07 -0.03 0.01 -0.03 0.08 0.02 0.06 -0.07
PLAY46 0.02 -0.11 0.11 0.81 0.01 0.05 0.02 0.01 -0.08 0.02 -0.01 -0.04 -0.02 0.00 -0.05 0.06
IMP47 0.03 -0.06 0.04 0.09 -0.04 -0.04 0.07 -0.03 0.03 0.10 0.65 0.04 0.06 0.06 -0.03 0.14
IMP48 -0.08 0.02 0.06 0.00 -0.01 0.04 0.08 0.00 0.04 0.04 0.73 0.01 -0.02 0.04 0.02 -0.08
MISC51 0.02 0.11 -0.01 -0.06 -0.03 0.23 0.04 0.15 -0.05 0.04 0.05 -0.05 0.12 0.27 -0.27 0.29
MISC52 -0.03 0.04 0.08 -0.02 0.03 -0.06 0.02 -0.03 0.08 0.02 0.00 0.01 -0.03 0.76 0.11 -0.07
MISC53 -0.04 -0.06 -0.02 -0.05 -0.02 0.03 -0.01 0.03 0.02 0.05 0.07 0.04 0.02 0.68 -0.08 0.07
MISC54 -0.01 0.16 0.00 -0.04 0.65 -0.02 0.06 -0.02 0.01 0.02 0.01 0.11 -0.09 0.06 0.05 0.01
MISC55 0.00 -0.07 0.01 0.02 0.87 0.05 0.00 0.03 0.00 0.04 -0.06 0.01 0.00 -0.04 0.02 0.07
MISC56 -0.08 0.02 0.07 -0.09 0.78 -0.02 -0.01 -0.04 0.06 -0.02 0.06 0.01 0.10 0.05 -0.10 -0.05
MISC59 0.07 0.10 -0.09 0.04 0.00 0.27 -0.04 0.09 -0.08 -0.06 0.43 0.02 0.32 0.03 -0.05 0.00
MISC60 0.00 0.01 -0.08 0.08 0.07 -0.01 0.07 -0.09 -0.01 0.04 0.08 0.04 0.66 0.01 0.09 0.03
MISC61 0.07 -0.01 -0.02 0.20 -0.01 0.12 0.11 0.06 -0.11 -0.02 0.03 -0.07 0.63 -0.03 0.03 -0.01
MISC62 -0.10 0.39 0.03 -0.04 0.00 -0.03 0.03 0.19 0.30 0.04 0.03 -0.01 0.21 0.03 0.03 0.02
MISC64 -0.13 0.77 0.08 -0.01 0.00 0.04 -0.01 -0.04 0.00 -0.02 -0.06 0.05 0.03 0.01 0.01 0.04
MISC65 -0.11 0.73 0.04 -0.03 -0.03 0.02 0.00 0.04 0.06 0.03 0.01 0.05 0.02 0.06 -0.01 0.07
MISC66 0.19 0.64 0.00 -0.04 0.16 0.06 -0.04 -0.05 0.00 0.02 0.09 0.01 -0.03 0.02 -0.03 -0.09
MISC67 0.18 0.61 0.00 -0.10 0.13 0.06 -0.03 0.03 0.00 -0.03 0.08 -0.03 0.00 0.03 0.06 -0.17
MISC69 0.09 0.04 0.00 -0.06 0.21 0.05 0.04 -0.02 0.05 -0.07 0.09 -0.03 0.01 -0.01 0.11 0.51
2.5.6.4.1 16 Factor Model Largest Loading Per Item
Largest Loading Per Item and Associated Factors, Sorted by Factor
largest maximum itemNames itemDescriptions
37 PA1 0.39 MISC62 Becomes highly excited and/or distracted when encountering unfamiliar dogs
38 PA1 0.77 MISC64 When working, is easily distracted or preoccupied by odors/engages in persistent sniffing of ground or objects
39 PA1 0.73 MISC65 Has difficulty shifting attention away from interesting or distracting stimuli (e.g., other dogs, odor, people, small animals, etc.)
40 PA1 0.64 MISC66 Is distracted or nervous in new, unfamiliar environments, has difficulty maintaining focus on work
41 PA1 0.61 MISC67 Is slow to recover after being distracted, startled, or frightened/takes a long time to resume work
54 PA1 0.64 TRAIN07 Unfocused; is easily distracted by interesting sights, sounds or smells
29 PA10 0.76 MISC52 Begs persistently for food when people are eating
30 PA10 0.68 MISC53 Steals food
20 PA11 0.37 FEAR26 When examined/treated by a veterinarian
23 PA11 0.83 FEAR29 When having nails trimmed, or feet touched/handled
24 PA11 0.67 FEAR30 When groomed or bathed
21 PA12 0.86 FEAR27 When approached directly by an unfamiliar dog
25 PA12 0.75 FEAR31 When barked, growled, or lunged at by an unfamiliar dog
35 PA13 0.66 MISC60 Playful, puppyish, boisterous
36 PA13 0.63 MISC61 Active, energetic, always on the go
9 PA14 0.54 ATT39 Displays a strong attachment for you or another familiar person
10 PA14 0.47 ATT41 Tends to nudge or paw you (or others) for attention
26 PA15 0.65 IMP47 Impulsive; doesn’t seem to think before s/he acts
27 PA15 0.73 IMP48 Becomes frustrated/impatient in a wide range of situations
34 PA15 0.43 MISC59 Hyperactive, restless, has trouble settling down
28 PA16 0.29 MISC51 Escapes or would escape from home, yard or kennel given the chance
42 PA16 0.51 MISC69 Chases/follows shadows, light spots, etc.
43 PA2 0.51 PLAY43 Eagerly engages in play with new/unfamiliar people
44 PA2 0.84 PLAY44 Highly toy focused; attention riveted on tug toy/balls when these are held by handler or other person
45 PA2 0.84 PLAY45 Eagerly initiates play sessions; brings objects/toys to you/the handler and retrieves them when thrown
46 PA2 0.81 PLAY46 Hunts persistently for thrown or hidden toys/objects, not easily distracted from this task
1 PA3 0.73 AGG09 Barks, growls, attempts to bite when approached directly by an unfamiliar person while being walked/exercised on a leash
2 PA3 0.80 AGG10 Barks, etc. when unfamiliar persons approach the dog when s/he is in his/her kennel
3 PA3 0.64 AGG14 When strangers walk past when the dog is in his/her home run or kennel
4 PA3 0.72 AGG15 When an unfamiliar person tries to touch or pet the dog
5 PA4 0.51 AGG16 When approached directly by an unfamiliar dog while being walked/exercised on a leash
6 PA4 0.55 AGG18 When barked, growled, or lunged at by another dog
7 PA4 0.86 AGG19 Towards other familiar dogs
8 PA4 0.67 AGG20 When approached while playing with/chewing a favorite toy, bone, object, etc. by another familiar dog
31 PA5 0.65 MISC54 Reluctant to/nervous about crossing grates or other unfamiliar surfaces
32 PA5 0.87 MISC55 Reluctant to/nervous about crossing shiny or slippery floors
33 PA5 0.78 MISC56 Nervous or frightened when ascending or descending some types of stairs
11 PA6 0.57 EXCITE35 Excitable when you first arrive home, or at the dog’s kennel, after a brief absence
12 PA6 0.64 EXCITE36 Excitable when playing with you or other familiar persons
13 PA6 0.83 EXCITE37 Excitable just before being taken out for a walk
14 PA6 0.77 EXCITE38 Excitable just before being taken out for work or training
47 PA7 0.79 SEPR32 Restlessness/agitation/pacing when left alone
48 PA7 0.79 SEPR33 Barking or whining when left alone
49 PA7 0.79 SEPR34 Chewing/scratching at doors, floor, fencing, etc., when left alone
50 PA8 0.43 TRAIN01 Is hard to recall when off the leash
51 PA8 0.64 TRAIN02 Is slow to obey a ‘sit’ command
52 PA8 0.76 TRAIN03 Is slow to obey a ‘stay’ command
53 PA8 0.45 TRAIN04 Has difficulty attending/listening to things you say or do
15 PA9 0.82 FEAR21 When an unfamiliar person approaches the dog when s/he is away from his/her normal home environment or kennel
16 PA9 0.36 FEAR22 In response to sudden or loud noises (e.g., gun fire, car backfire, road drills, objects being dropped, etc.)
17 PA9 0.77 FEAR23 When an unfamiliar person visits your home or approaches the dog when in his/her home kennel
18 PA9 0.81 FEAR24 When an unfamiliar person tries to touch or pet the dog
19 PA9 0.41 FEAR25 In response to strange or unfamiliar objects on or near sidewalks or walkways (e.g., plastic trash bags, leaves, litter, flags flapping, etc.
22 PA9 0.49 FEAR28 When first exposed to unfamiliar situations (e.g., novel environments, first visit to the veterinarian, etc.)

2.5.7 16 Model Reliability Measures

PA9 PA1 PA3 PA2 PA5 PA7 PA6 PA4 PA12 PA8 PA15 PA11 PA13 PA10 PA14 PA16
alpha 0.777 0.786 0.76 0.812 0.718 0.735 0.787 0.753 0.802 0.693 0.683 0.613 0.71 0.622 0.439 0.159

omega is thought to be better measure of reliability than alpha, because it looks at the relationships between all the items rather than just the ones within each factor. It’s different than alpha because it assumes that there is a genral factor, g, that all items have in common. This is listed as the first row of the table below.

There are three omegas for each factor:

  • omega_total - total true score variance in factor
  • omega_hierarchical - variance attributable to general factor (g)
  • omega_subscale - variance attributable to the subscale

From EFAtools::OMEGA documentation:

Omegas refer to the correlation between a factor and a unit-weighted composite score and thus the true score variance in a unit-weighted composite based on the respective indicators. Omega total is the total true score variance in a composite. Omega hierarchical is the true score variance in a composite that is attributable to the general factor, and omega subscale is the true score variance in a composite attributable to all subscales / group factors (for the whole scale) or to the specific subscale / group factor (for subscale composites).

Interpretation of omega_subscale

no universal guideline

0.50 is minimum, 0.75 preferable (watkins books,rodrigues16, zimbarg2005

## Omega total, omega hierarchical, omega subscale, H index, explained common variance (ECV), and percent of uncontaminated correlations (PUC) for the general factor (top row) and omegas and H index for the group factors:
## 
##       tot  hier   sub     H  ECV   PUC
## g   0.948 0.590 0.184 0.884 0.23 0.946
## F1  0.847 0.389 0.458 0.710           
## F2  0.701 0.090 0.611 0.648           
## F3  0.720 0.267 0.454 0.631           
## F4  0.820 0.199 0.621 0.726           
## F5  0.521 0.003 0.518 0.590           
## F6  0.410 0.028 0.381 0.408           
## F7  0.630 0.188 0.442 0.591           
## F8  0.403 0.138 0.265 0.305           
## F9  0.811 0.055 0.756 0.862           
## F10 0.797 0.105 0.692 0.809           
## F11 0.701 0.151 0.549 0.761           
## F12 0.840 0.185 0.655 0.769           
## F13 0.773 0.030 0.743 0.823           
## F14 0.855 0.181 0.673 0.759           
## F15 0.673 0.205 0.468 0.647           
## F16 0.809 0.292 0.517 0.804

2.6 17 Factors

2.6.1 Imputation of Missing Items (17 factors)

include reference on why imputation is better than removing observations, pairwise removing is biased, using means isn’t good.

Categorical missing values using multiple Correspondence Analysis (also called Missing Fuzzy Average method) Josseet al (201 0)

2.6.2 17-Factors: Pre-Imputation Tests for Data Suitability for Factor Analysis

2.6.2.1 Bartlett’s Test of Sphericity

This function tests whether a correlation matrix is significantly different from an identity matrix (Bartlett, 1951). If the Bartlett’s test is not significant, the correlation matrix is not suitable for factor analysis because the variables show too little covariance.

## Warning in polychoric(ddNum17, smooth = TRUE, correct = 0.01): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1322 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 4.0773023^{4} with 1540 DF, p = 0e+00.

2.6.2.2 Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8532378

2.6.3 17-Factor: Post-Imputation Tests for Data Suitability for Factor Analysis

2.6.3.1 Bartlett’s Test of Sphericity

## Warning in polychoric(imputedNumericDF17, smooth = TRUE, correct = 0.01): The
## items do not have an equal number of response alternatives, global set to
## FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1321 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.9253542^{4} with 1540 DF, p = 0e+00.

2.6.3.2 17-Factor: Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8532378

2.6.4 Between-Item Correlations

2.6.4.1 Pre-Imputation

For factor analysis, it is recommended that some of the item correlationsshould be between 0.3 and 0.9. Polychoric correlations were computed using the polychoric function in the psych package in R with options smooth = TRUE to adjust for non-positive-definite matrices and correct = 0.01 to avoid computation problems with covariance of zero.

The minimum correlation in this data set is -0.477. The maximum correlation in this data set is 0.856.

#### Post-Imputation

The post imputation polyserial correlation was also computed using the polychoric function, with no setting smooth = TRUE and correct = 0.01.

The minimum correlation was -0.466. The maximum correlation was 0.851.

2.6.5 Estimating Number of Factors

2.6.5.1 Parallel method

From the EFAtools documentation:

Various methods for performing parallel analysis. This function uses future_lapply for which a parallel processing plan can be selected. To do so, call library(future) and, for example, plan(multisession); see examples.

Settings Used:
- n.obs = 1117 - eigen_type = “EFA” - fa = “fa” (factor analysis not PCA) - fm = “wls” (weighted least squares because pa gave errors and weighted recommended for ordinal data. - use = “all.obs” (since we have imputed missing values, we can use all data points) - cor = “poly” (use polychoric correlation matrix) - n.iter = 100 (run for 100 iterations)

The suggested number of factors is 13.

2.6.5.2 Minimum Average Partial

MAP is recommended as a way to find the number of factors when the items are ordinal. The lowest value indicates the best number of factors.

Number of Factors MAP value
1 0.0271
2 0.0232
3 0.0182
4 0.0160
5 0.0146
6 0.0142
7 0.0141
8 0.0132
9 0.0130
10 0.0127
11 0.0123
12 0.0127
13 0.0127
14 0.0127
15 0.0132
16 0.0138
17 0.0145
18 0.0149

2.6.6 Factor Analysis for 17 Factors

2.6.6.1 17 Factors Model Fit

## Warning in polychoric(r, correct = correct, weight = weight): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1321 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

Although the chi-square test of goodness of fit is sensitive to departures from normality like the C-BARQ items, Hopper et al (2008) recommend always reporting it.

  • chi-square: 5699.9605713
  • degrees of freedom: 724
  • P-value for chi-square = 0e+00

Tucker-Lewis Index of Factoring Reliability/Non-Norm Fit Index: 0.7162758. Should be > 0.9; need reference)

2.6.6.2 17 Factor Model Communalities

fanal17.communality
TRAIN01 0.53
TRAIN02 0.47
TRAIN03 0.58
TRAIN04 0.59
TRAIN07 0.61
AGG09 0.83
AGG10 0.70
AGG14 0.52
AGG15 0.81
AGG16 0.76
AGG18 0.74
AGG19 0.77
AGG20 0.58
FEAR21 0.85
FEAR22 0.47
FEAR23 0.81
FEAR24 0.87
FEAR25 0.61
FEAR26 0.45
FEAR27 0.89
FEAR28 0.67
FEAR29 0.53
FEAR30 0.65
FEAR31 0.66
SEPR32 0.67
SEPR33 0.66
SEPR34 0.71
EXCITE35 0.48
EXCITE36 0.57
EXCITE37 0.66
EXCITE38 0.68
ATT39 0.63
ATT40 0.49
ATT41 0.32
PLAY43 0.57
PLAY44 0.82
PLAY45 0.76
PLAY46 0.74
IMP47 0.60
IMP48 0.61
MISC51 0.54
MISC52 0.61
MISC53 0.54
MISC54 0.62
MISC55 0.76
MISC56 0.64
MISC59 0.58
MISC60 0.58
MISC61 0.70
MISC62 0.57
MISC63 0.53
MISC64 0.64
MISC65 0.64
MISC66 0.69
MISC67 0.62
MISC69 0.23

2.6.6.3 How many communalities < 0.40?

There are 2 items with communality < 0.40.

fanal17.communality item
ATT41 0.32 ATT41
MISC69 0.23 MISC69

2.6.6.4 17 Factor Model Loadings

PA16 PA1 PA3 PA2 PA7 PA5 PA6 PA4 PA9 PA8 PA11 PA15 PA14 PA10 PA12 PA17 PA13
TRAIN01 -0.02 0.15 -0.04 -0.02 -0.09 0.00 -0.06 0.01 -0.03 0.44 0.08 -0.01 -0.02 0.08 -0.07 0.06 0.35
TRAIN02 -0.03 0.05 0.07 0.00 0.05 0.01 0.03 -0.08 0.06 0.65 0.03 0.03 0.02 0.03 -0.02 -0.06 -0.15
TRAIN03 -0.03 -0.05 -0.03 -0.05 0.03 0.05 0.02 0.03 -0.07 0.72 0.05 0.04 0.05 0.01 0.05 0.01 0.02
TRAIN04 0.07 0.32 -0.04 -0.02 -0.02 0.02 0.04 -0.05 0.02 0.47 -0.04 0.16 -0.07 -0.04 -0.15 0.00 0.13
TRAIN07 0.07 0.61 -0.10 -0.05 -0.06 0.03 -0.02 0.02 0.01 0.23 -0.01 0.01 -0.05 -0.02 0.01 0.07 0.03
AGG09 0.21 -0.02 0.73 0.02 0.06 -0.06 -0.04 -0.03 -0.04 0.01 0.07 0.06 0.01 -0.13 0.11 0.10 0.17
AGG10 -0.05 -0.01 0.82 0.07 -0.03 0.13 0.00 0.06 0.02 0.01 0.01 0.02 -0.05 0.03 0.02 -0.03 -0.09
AGG14 0.05 0.08 0.64 0.08 -0.01 0.15 0.04 0.07 0.00 -0.03 -0.10 -0.02 -0.14 0.15 0.04 0.07 -0.02
AGG15 0.28 0.04 0.74 -0.07 0.05 -0.05 0.00 0.12 -0.07 0.00 0.07 -0.01 0.10 0.04 -0.11 -0.08 0.00
AGG16 -0.15 -0.01 0.29 -0.06 0.05 -0.05 -0.04 0.47 0.33 0.11 -0.05 0.01 -0.02 -0.17 0.14 0.11 0.04
AGG18 -0.16 -0.08 0.27 -0.03 -0.01 -0.03 -0.11 0.51 0.28 0.07 -0.02 0.06 0.07 -0.11 0.15 0.12 0.07
AGG19 0.04 0.04 0.04 0.06 0.01 0.00 0.10 0.82 0.02 -0.01 0.08 -0.09 0.06 -0.04 -0.06 0.02 0.06
AGG20 0.02 0.03 -0.06 -0.02 0.02 0.01 0.00 0.74 -0.06 -0.05 0.06 0.17 -0.11 0.11 -0.02 -0.07 -0.07
FEAR21 0.82 0.01 0.09 0.00 0.03 -0.01 0.00 0.02 0.09 -0.02 0.08 0.02 0.01 -0.03 0.05 -0.02 0.04
FEAR22 0.30 0.07 -0.20 -0.04 0.14 0.07 0.00 0.20 -0.10 0.08 0.02 -0.05 -0.07 0.11 0.02 0.30 -0.24
FEAR23 0.74 -0.03 0.17 -0.04 0.03 0.07 -0.02 -0.02 0.15 -0.01 -0.03 0.00 0.04 -0.05 0.00 0.10 -0.02
FEAR24 0.79 0.03 0.13 -0.10 -0.02 -0.03 0.04 -0.01 0.17 -0.01 0.04 -0.01 0.02 -0.03 0.07 -0.03 -0.03
FEAR25 0.36 0.05 -0.10 0.06 0.20 0.08 -0.02 0.08 -0.03 0.08 -0.01 0.03 -0.13 0.08 0.02 0.45 -0.04
FEAR26 0.34 0.02 0.03 0.05 -0.02 0.04 -0.02 0.09 0.09 0.05 0.42 0.02 0.02 0.04 0.02 -0.08 -0.08
FEAR27 0.16 0.02 -0.04 0.02 0.04 0.03 0.01 0.04 0.88 -0.02 0.01 0.01 -0.03 0.04 -0.02 -0.01 0.00
FEAR28 0.43 0.06 -0.06 0.03 0.30 0.05 0.04 0.00 0.03 0.04 0.18 0.05 -0.06 -0.10 0.00 0.18 -0.14
FEAR29 -0.02 -0.02 0.02 0.02 -0.02 0.03 0.01 0.00 0.03 0.06 0.72 -0.05 0.00 0.04 0.04 -0.04 0.04
FEAR30 -0.03 -0.01 -0.01 0.02 0.06 0.09 -0.01 0.07 0.00 0.00 0.74 0.02 -0.04 0.04 -0.04 0.05 -0.03
FEAR31 0.11 0.02 -0.12 0.00 0.06 0.03 0.01 -0.02 0.73 0.01 0.10 0.04 -0.06 0.05 0.01 0.02 -0.03
SEPR32 0.04 0.00 0.04 -0.08 -0.06 0.75 0.08 0.03 0.01 -0.03 0.07 0.05 0.02 -0.12 0.07 -0.02 -0.06
SEPR33 -0.08 0.04 0.03 0.00 0.01 0.78 0.00 -0.03 0.05 0.07 0.04 0.00 0.03 0.01 0.02 -0.01 -0.07
SEPR34 -0.01 0.00 0.04 0.01 0.06 0.80 -0.02 -0.04 0.01 0.01 0.02 -0.01 0.02 0.04 -0.03 0.01 0.16
EXCITE35 0.12 0.04 -0.03 -0.08 -0.07 0.12 0.56 0.04 0.02 0.08 -0.06 -0.04 0.14 0.04 0.13 -0.05 -0.02
EXCITE36 0.01 0.00 0.00 0.14 0.02 -0.05 0.63 0.00 -0.08 0.01 0.06 0.04 0.13 -0.05 0.06 0.01 -0.05
EXCITE37 -0.02 0.04 0.03 -0.04 0.01 0.01 0.83 0.02 0.04 -0.01 -0.02 0.01 -0.02 -0.01 0.01 -0.08 0.05
EXCITE38 -0.02 -0.13 -0.01 0.05 0.03 0.00 0.78 0.02 0.00 0.01 0.01 0.06 -0.06 0.02 -0.03 0.12 -0.01
ATT39 0.05 0.03 -0.01 0.06 -0.03 0.01 0.07 -0.03 -0.02 0.00 -0.02 -0.09 0.09 0.08 0.75 -0.09 -0.06
ATT40 0.02 0.15 -0.03 -0.13 0.05 0.07 0.04 -0.08 -0.04 -0.08 0.12 0.17 -0.17 -0.08 0.50 0.10 0.15
ATT41 0.02 -0.01 -0.13 0.05 -0.02 0.09 -0.01 0.06 -0.04 0.02 -0.02 0.12 0.06 0.20 0.40 0.14 0.03
PLAY43 -0.29 -0.01 -0.22 0.49 0.00 -0.03 0.08 -0.08 0.09 -0.05 -0.02 0.01 0.10 0.05 -0.07 0.09 -0.05
PLAY44 -0.02 0.05 -0.02 0.86 -0.02 -0.03 0.06 0.08 0.01 -0.02 0.06 0.06 0.03 -0.11 0.03 -0.02 0.02
PLAY45 -0.01 -0.01 0.06 0.81 -0.05 -0.06 -0.02 -0.07 0.06 -0.05 0.00 0.03 0.10 0.00 0.05 0.02 -0.07
PLAY46 0.03 -0.09 0.10 0.82 0.02 0.05 0.02 0.02 -0.07 0.02 -0.05 -0.01 -0.02 0.01 -0.04 -0.02 0.05
IMP47 0.05 -0.05 0.00 0.16 0.00 -0.02 0.08 0.00 0.05 0.14 -0.02 0.61 0.04 0.09 -0.01 -0.02 0.14
IMP48 -0.08 0.00 0.07 0.02 -0.01 0.04 0.09 0.01 0.04 0.06 0.02 0.71 0.01 0.05 -0.03 0.02 -0.09
MISC51 0.05 0.08 -0.05 -0.04 0.01 0.23 0.05 0.12 -0.07 0.03 -0.04 0.07 0.04 0.19 -0.08 0.03 0.52
MISC52 -0.05 0.02 0.13 -0.09 0.01 -0.06 0.01 -0.04 0.09 0.02 0.06 0.04 0.01 0.72 0.09 0.07 -0.05
MISC53 -0.04 -0.02 -0.02 -0.08 0.01 0.01 0.01 0.04 0.03 0.04 0.08 0.11 0.01 0.61 0.01 -0.06 0.20
MISC54 -0.01 0.13 -0.01 -0.03 0.65 -0.03 0.05 -0.02 0.01 0.02 0.11 0.01 -0.10 0.05 0.05 0.06 -0.02
MISC55 -0.01 -0.04 0.00 0.04 0.88 0.05 0.00 0.04 0.01 0.03 -0.03 -0.08 0.01 -0.02 -0.01 0.01 -0.01
MISC56 -0.07 0.04 0.05 -0.09 0.80 -0.06 -0.01 -0.04 0.05 -0.03 0.03 0.08 0.08 0.00 -0.03 -0.08 0.04
MISC59 0.07 0.10 -0.10 0.05 0.01 0.26 -0.04 0.10 -0.08 -0.05 0.03 0.44 0.32 0.01 -0.01 -0.01 0.04
MISC60 0.01 -0.03 -0.07 0.07 0.06 0.02 0.05 -0.09 -0.01 0.06 0.01 0.06 0.67 0.03 0.07 0.05 0.01
MISC61 0.08 -0.01 -0.01 0.18 -0.01 0.12 0.08 0.07 -0.11 -0.01 -0.08 0.03 0.66 -0.02 0.03 -0.01 -0.01
MISC62 -0.16 0.26 0.08 -0.12 -0.07 -0.03 0.03 0.09 0.29 0.01 0.07 0.02 0.25 -0.04 0.00 0.35 0.07
MISC63 -0.09 0.21 0.11 -0.09 -0.02 -0.06 0.16 -0.10 0.05 -0.04 0.12 0.06 0.18 0.07 -0.08 0.50 0.09
MISC64 -0.09 0.82 0.05 0.03 0.04 0.03 -0.02 0.01 0.01 -0.01 0.00 -0.07 0.02 0.03 0.03 -0.05 0.02
MISC65 -0.08 0.70 0.02 -0.02 0.00 0.02 -0.02 0.06 0.06 0.05 0.03 0.01 0.01 0.05 0.01 0.05 0.07
MISC66 0.18 0.57 0.01 -0.07 0.15 0.04 -0.04 -0.07 -0.02 0.02 0.06 0.12 -0.04 -0.04 0.02 0.11 -0.04
MISC67 0.16 0.56 0.01 -0.12 0.12 0.03 -0.04 0.03 -0.02 -0.03 0.02 0.12 0.00 -0.02 0.08 0.08 -0.15
MISC69 0.10 0.04 -0.05 0.04 0.24 0.12 0.02 0.04 0.09 -0.01 -0.18 0.04 -0.03 0.08 0.08 0.09 0.24
2.6.6.4.1 17 Factor Model Largest Loading Per Item
Largest Loading Per Item and Associated Factors, Sorted by Factor
largest maximum itemNames itemDescriptions
40 PA1 0.82 MISC64 When working, is easily distracted or preoccupied by odors/engages in persistent sniffing of ground or objects
41 PA1 0.70 MISC65 Has difficulty shifting attention away from interesting or distracting stimuli (e.g., other dogs, odor, people, small animals, etc.)
42 PA1 0.57 MISC66 Is distracted or nervous in new, unfamiliar environments, has difficulty maintaining focus on work
43 PA1 0.56 MISC67 Is slow to recover after being distracted, startled, or frightened/takes a long time to resume work
56 PA1 0.61 TRAIN07 Unfocused; is easily distracted by interesting sights, sounds or smells
30 PA10 0.72 MISC52 Begs persistently for food when people are eating
31 PA10 0.61 MISC53 Steals food
21 PA11 0.42 FEAR26 When examined/treated by a veterinarian
24 PA11 0.72 FEAR29 When having nails trimmed, or feet touched/handled
25 PA11 0.74 FEAR30 When groomed or bathed
9 PA12 0.75 ATT39 Displays a strong attachment for you or another familiar person
10 PA12 0.50 ATT40 Unwilling to leave your side, even when working, reluctant to work at a distance from you/the handler
11 PA12 0.40 ATT41 Tends to nudge or paw you (or others) for attention
29 PA13 0.52 MISC51 Escapes or would escape from home, yard or kennel given the chance
44 PA13 0.24 MISC69 Chases/follows shadows, light spots, etc.
36 PA14 0.67 MISC60 Playful, puppyish, boisterous
37 PA14 0.66 MISC61 Active, energetic, always on the go
27 PA15 0.61 IMP47 Impulsive; doesn’t seem to think before s/he acts
28 PA15 0.71 IMP48 Becomes frustrated/impatient in a wide range of situations
35 PA15 0.44 MISC59 Hyperactive, restless, has trouble settling down
16 PA16 0.82 FEAR21 When an unfamiliar person approaches the dog when s/he is away from his/her normal home environment or kennel
17 PA16 0.30 FEAR22 In response to sudden or loud noises (e.g., gun fire, car backfire, road drills, objects being dropped, etc.)
18 PA16 0.74 FEAR23 When an unfamiliar person visits your home or approaches the dog when in his/her home kennel
19 PA16 0.79 FEAR24 When an unfamiliar person tries to touch or pet the dog
23 PA16 0.43 FEAR28 When first exposed to unfamiliar situations (e.g., novel environments, first visit to the veterinarian, etc.)
20 PA17 0.45 FEAR25 In response to strange or unfamiliar objects on or near sidewalks or walkways (e.g., plastic trash bags, leaves, litter, flags flapping, etc.
38 PA17 0.35 MISC62 Becomes highly excited and/or distracted when encountering unfamiliar dogs
39 PA17 0.50 MISC63 Becomes highly excited and/or distracted when encountering unfamiliar people
45 PA2 0.49 PLAY43 Eagerly engages in play with new/unfamiliar people
46 PA2 0.86 PLAY44 Highly toy focused; attention riveted on tug toy/balls when these are held by handler or other person
47 PA2 0.81 PLAY45 Eagerly initiates play sessions; brings objects/toys to you/the handler and retrieves them when thrown
48 PA2 0.82 PLAY46 Hunts persistently for thrown or hidden toys/objects, not easily distracted from this task
1 PA3 0.73 AGG09 Barks, growls, attempts to bite when approached directly by an unfamiliar person while being walked/exercised on a leash
2 PA3 0.82 AGG10 Barks, etc. when unfamiliar persons approach the dog when s/he is in his/her kennel
3 PA3 0.64 AGG14 When strangers walk past when the dog is in his/her home run or kennel
4 PA3 0.74 AGG15 When an unfamiliar person tries to touch or pet the dog
5 PA4 0.47 AGG16 When approached directly by an unfamiliar dog while being walked/exercised on a leash
6 PA4 0.51 AGG18 When barked, growled, or lunged at by another dog
7 PA4 0.82 AGG19 Towards other familiar dogs
8 PA4 0.74 AGG20 When approached while playing with/chewing a favorite toy, bone, object, etc. by another familiar dog
49 PA5 0.75 SEPR32 Restlessness/agitation/pacing when left alone
50 PA5 0.78 SEPR33 Barking or whining when left alone
51 PA5 0.80 SEPR34 Chewing/scratching at doors, floor, fencing, etc., when left alone
12 PA6 0.56 EXCITE35 Excitable when you first arrive home, or at the dog’s kennel, after a brief absence
13 PA6 0.63 EXCITE36 Excitable when playing with you or other familiar persons
14 PA6 0.83 EXCITE37 Excitable just before being taken out for a walk
15 PA6 0.78 EXCITE38 Excitable just before being taken out for work or training
32 PA7 0.65 MISC54 Reluctant to/nervous about crossing grates or other unfamiliar surfaces
33 PA7 0.88 MISC55 Reluctant to/nervous about crossing shiny or slippery floors
34 PA7 0.80 MISC56 Nervous or frightened when ascending or descending some types of stairs
52 PA8 0.44 TRAIN01 Is hard to recall when off the leash
53 PA8 0.65 TRAIN02 Is slow to obey a ‘sit’ command
54 PA8 0.72 TRAIN03 Is slow to obey a ‘stay’ command
55 PA8 0.47 TRAIN04 Has difficulty attending/listening to things you say or do
22 PA9 0.88 FEAR27 When approached directly by an unfamiliar dog
26 PA9 0.73 FEAR31 When barked, growled, or lunged at by an unfamiliar dog

2.6.7 17 Model Reliability Measures

PA16 PA1 PA3 PA2 PA7 PA5 PA6 PA4 PA9 PA8 PA11 PA15 PA14 PA10 PA12 PA17 PA13
alpha 0.761 0.805 0.76 0.812 0.718 0.735 0.787 0.752 0.802 0.693 0.613 0.683 0.71 0.622 0.494 0.558 0.159

omega is thought to be better measure of reliability than alpha, because it looks at the relationships between all the items rather than just the ones within each factor. It’s different than alpha because it assumes that there is a genral factor, g, that all items have in common. This is listed as the first row of the table below.

There are three omegas for each factor:

  • omega_total - total true score variance in factor
  • omega_hierarchical - variance attributable to general factor (g)
  • omega_subscale - variance attributable to the subscale

From EFAtools::OMEGA documentation:

Omegas refer to the correlation between a factor and a unit-weighted composite score and thus the true score variance in a unit-weighted composite based on the respective indicators. Omega total is the total true score variance in a composite. Omega hierarchical is the true score variance in a composite that is attributable to the general factor, and omega subscale is the true score variance in a composite attributable to all subscales / group factors (for the whole scale) or to the specific subscale / group factor (for subscale composites).

Interpretation of omega_subscale

no universal guideline

0.50 is minimum, 0.75 preferable (watkins books,rodrigues16, zimbarg2005

## Omega total, omega hierarchical, omega subscale, H index, explained common variance (ECV), and percent of uncontaminated correlations (PUC) for the general factor (top row) and omegas and H index for the group factors:
## 
##       tot  hier   sub     H   ECV   PUC
## g   0.950 0.600 0.162 0.893 0.242 0.953
## F1  0.832 0.400 0.432 0.683            
## F2  0.631 0.076 0.555 0.612            
## F3  0.705 0.253 0.452 0.596            
## F4  0.633 0.090 0.543 0.627            
## F5  0.365 0.128 0.237 0.293            
## F6  0.556 0.002 0.555 0.619            
## F7  0.588 0.162 0.426 0.571            
## F8  0.792 0.289 0.503 0.785            
## F9  0.631 0.371 0.260 0.358            
## F10 0.797 0.063 0.734 0.856            
## F11 0.812 0.110 0.702 0.815            
## F12 0.689 0.156 0.534 0.750            
## F13 0.841 0.156 0.685 0.767            
## F14 0.760 0.025 0.736 0.820            
## F15 0.851 0.207 0.644 0.763            
## F16 0.661 0.184 0.477 0.642            
## F17 0.826 0.213 0.613 0.730

2.7 18 Factors

2.7.1 Imputation of Missing Items (18 factors)

include reference on why imputation is better than removing observations, pairwise removing is biased, using means isn’t good.

Categorical missing values using multiple Correspondence Analysis (also called Missing Fuzzy Average method) Josseet al (201 0)

2.7.2 18-Factors: Pre-Imputation Tests for Data Suitability for Factor Analysis

2.7.2.1 Bartlett’s Test of Sphericity

This function tests whether a correlation matrix is significantly different from an identity matrix (Bartlett, 1951). If the Bartlett’s test is not significant, the correlation matrix is not suitable for factor analysis because the variables show too little covariance.

## Warning in polychoric(ddNum18, smooth = TRUE, correct = 0.01): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1318 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 4.0386817^{4} with 1540 DF, p = 0e+00.

2.7.2.2 Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8554013

2.7.3 18-Factor: Post-Imputation Tests for Data Suitability for Factor Analysis

2.7.3.1 Bartlett’s Test of Sphericity

## Warning in polychoric(imputedNumericDF18, smooth = TRUE, correct = 0.01): The
## items do not have an equal number of response alternatives, global set to
## FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1317 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

This is a test that the matrix is an identity matrix. This would mean that the correlations were not significantly different from 0. If it’s not significant, the matrix is not suitable because the variables show too little covariance.

The chi-squared for the Bartlett test is 3.8935509^{4} with 1540 DF, p = 0e+00.

2.7.3.2 18-Factor: Kaiser-Meyer-Olkin Criterion (KMO)

From EFAtools::KMO documentation:

The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis

The numeric version of the dataset is used because stats::cor is used to find the correlation and it requires numeric input. Used option for Spearman correlation because of ordered variables.

## ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.

[1] 0.8554013

2.7.4 Between-Item Correlations

2.7.4.1 Pre-Imputation

For factor analysis, it is recommended that some of the item correlationsshould be between 0.3 and 0.9. Polychoric correlations were computed using the polychoric function in the psych package in R with options smooth = TRUE to adjust for non-positive-definite matrices and correct = 0.01 to avoid computation problems with covariance of zero.

The minimum correlation in this data set is -0.477. The maximum correlation in this data set is 0.856.

#### Post-Imputation

The post imputation polyserial correlation was also computed using the polychoric function, with no setting smooth = TRUE and correct = 0.01.

The minimum correlation was -0.466. The maximum correlation was 0.851.

2.7.5 Estimating Number of Factors

2.7.5.1 Parallel method

From the EFAtools documentation:

Various methods for performing parallel analysis. This function uses future_lapply for which a parallel processing plan can be selected. To do so, call library(future) and, for example, plan(multisession); see examples.

Settings Used:
- n.obs = 1117 - eigen_type = “EFA” - fa = “fa” (factor analysis not PCA) - fm = “wls” (weighted least squares because pa gave errors and weighted recommended for ordinal data. - use = “all.obs” (since we have imputed missing values, we can use all data points) - cor = “poly” (use polychoric correlation matrix) - n.iter = 100 (run for 100 iterations)

The suggested number of factors is 13.

2.7.5.2 Minimum Average Partial

MAP is recommended as a way to find the number of factors when the items are ordinal. The lowest value indicates the best number of factors.

Number of Factors MAP value
1 0.0269
2 0.0229
3 0.0180
4 0.0158
5 0.0148
6 0.0144
7 0.0141
8 0.0130
9 0.0127
10 0.0122
11 0.0121
12 0.0124
13 0.0125
14 0.0125
15 0.0129
16 0.0135
17 0.0140
18 0.0148

2.7.6 Factor Analysis for 18 Factors

2.7.6.1 18 Factors Model Fit

## Warning in polychoric(r, correct = correct, weight = weight): The items do not
## have an equal number of response alternatives, global set to FALSE.
## Warning in matpLower(x, nvar, gminx, gmaxx, gminy, gmaxy): 1317 cells were
## adjusted for 0 values using the correction for continuity. Examine your data
## carefully.

Although the chi-square test of goodness of fit is sensitive to departures from normality like the C-BARQ items, Hopper et al (2008) recommend always reporting it.

  • chi-square: 5064.6026591
  • degrees of freedom: 685
  • P-value for chi-square = 0e+00

Tucker-Lewis Index of Factoring Reliability/Non-Norm Fit Index: 0.733559. Should be > 0.9; need reference)

2.7.6.2 18 Factor Model Communalities

fanal18.communality
TRAIN01 0.44
TRAIN02 0.43
TRAIN03 0.60
TRAIN04 0.59
TRAIN07 0.62
AGG09 0.80
AGG10 0.73
AGG14 0.58
AGG15 0.80
AGG16 0.76
AGG18 0.74
AGG19 0.77
AGG20 0.58
FEAR21 0.84
FEAR22 0.50
FEAR23 0.81
FEAR24 0.92
FEAR25 0.62
FEAR26 0.44
FEAR27 0.86
FEAR28 0.67
FEAR29 0.67
FEAR30 0.61
FEAR31 0.73
SEPR32 0.71
SEPR33 0.65
SEPR34 0.66
EXCITE35 0.48
EXCITE36 0.57
EXCITE37 0.66
EXCITE38 0.69
ATT39 0.61
ATT40 0.48
ATT41 0.32
PLAY43 0.57
PLAY44 0.83
PLAY45 0.78
PLAY46 0.73
IMP47 0.59
IMP48 0.60
IMP49 0.38
MISC52 0.60
MISC53 0.64
MISC54 0.62
MISC55 0.75
MISC56 0.67
MISC59 0.58
MISC60 0.61
MISC61 0.73
MISC62 0.62
MISC63 0.57
MISC64 0.64
MISC65 0.64
MISC66 0.70
MISC67 0.67
MISC69 0.53

2.7.6.3 How many communalities < 0.40?

There are 2 items with communality < 0.40.

fanal18.communality item
ATT41 0.32 ATT41
IMP49 0.38 IMP49

2.7.6.4 18 Factor Model Loadings

PA18 PA1 PA2 PA3 PA7 PA5 PA6 PA4 PA9 PA8 PA15 PA13 PA11 PA17 PA10 PA14 PA12 PA16
TRAIN01 -0.07 0.16 -0.07 0.00 -0.11 0.01 -0.07 0.00 -0.04 0.46 0.06 -0.02 0.08 0.10 0.07 -0.10 -0.03 0.08
TRAIN02 -0.01 0.05 0.02 0.05 0.07 0.01 0.03 -0.07 0.08 0.62 0.01 0.01 0.02 -0.07 0.03 -0.01 0.03 -0.10
TRAIN03 -0.02 -0.05 -0.03 -0.06 0.04 0.07 0.01 0.05 -0.08 0.74 0.02 0.03 0.02 0.00 0.05 0.05 0.02 -0.02
TRAIN04 0.05 0.32 -0.05 -0.03 -0.04 0.02 0.04 -0.06 0.02 0.47 0.16 -0.04 -0.01 0.01 -0.04 -0.17 -0.01 0.12
TRAIN07 0.04 0.59 -0.07 -0.07 -0.08 0.02 -0.02 0.01 0.01 0.23 0.00 -0.03 0.02 0.05 -0.03 0.01 0.07 0.10
AGG09 0.30 -0.04 0.02 0.61 0.07 -0.05 -0.06 0.01 -0.10 0.04 0.08 -0.01 0.06 0.19 -0.11 0.10 -0.12 0.05
AGG10 -0.06 -0.01 0.04 0.83 -0.02 0.09 0.00 0.05 0.05 0.01 0.02 0.00 0.03 -0.03 -0.02 0.02 -0.02 -0.07
AGG14 -0.02 0.09 0.01 0.73 -0.06 0.07 0.06 0.02 0.04 -0.06 -0.02 -0.04 -0.01 -0.03 0.06 0.02 0.12 0.14
AGG15 0.39 0.03 -0.05 0.60 0.08 -0.04 -0.01 0.15 -0.11 0.00 -0.02 0.06 0.04 0.03 0.06 -0.11 -0.15 -0.07
AGG16 -0.07 -0.04 -0.03 0.21 0.06 -0.03 -0.05 0.53 0.25 0.11 0.01 -0.07 -0.06 0.18 -0.13 0.14 -0.09 0.04
AGG18 -0.09 -0.09 0.00 0.20 0.00 -0.01 -0.12 0.56 0.22 0.08 0.06 0.02 -0.04 0.19 -0.08 0.14 -0.07 0.03
AGG19 0.07 0.03 0.08 0.00 0.02 0.03 0.09 0.84 -0.02 0.01 -0.08 0.01 0.05 0.05 0.00 -0.07 0.00 0.00
AGG20 -0.05 0.04 -0.06 0.00 0.00 -0.01 0.01 0.71 -0.02 -0.07 0.17 -0.05 0.09 -0.17 0.08 -0.03 0.14 0.01
FEAR21 0.82 0.02 -0.01 0.06 0.02 -0.01 0.00 0.00 0.08 -0.01 0.03 0.03 0.08 -0.06 -0.03 0.03 0.07 0.05
FEAR22 0.14 0.06 -0.09 -0.07 0.11 0.02 0.01 0.15 0.00 0.06 -0.05 0.02 0.03 0.00 0.02 0.02 0.55 -0.04
FEAR23 0.76 -0.04 -0.02 0.12 0.02 0.07 -0.03 -0.03 0.12 0.00 0.01 0.02 -0.04 0.06 -0.05 -0.01 0.11 0.03
FEAR24 0.90 0.01 -0.03 0.00 0.00 0.00 0.03 0.00 0.10 -0.01 -0.03 -0.04 0.00 0.01 0.03 0.06 -0.02 -0.02
FEAR25 0.23 0.02 0.01 -0.01 0.13 0.04 -0.02 0.03 0.00 0.06 0.06 -0.07 0.04 0.20 -0.01 0.01 0.49 0.17
FEAR26 0.39 0.02 0.08 -0.03 0.01 0.06 -0.02 0.09 0.08 0.03 0.03 -0.01 0.36 -0.05 0.06 0.02 -0.01 -0.09
FEAR27 0.16 0.01 0.01 0.00 0.02 0.01 0.01 0.05 0.84 -0.03 0.00 -0.02 0.02 0.03 0.02 -0.03 -0.04 0.06
FEAR28 0.39 0.04 0.04 -0.05 0.30 0.06 0.04 -0.02 0.06 0.04 0.05 -0.05 0.13 0.06 -0.12 0.01 0.27 -0.07
FEAR29 -0.06 -0.01 -0.05 0.06 -0.04 -0.02 0.00 -0.03 0.05 0.03 -0.03 0.05 0.84 -0.02 -0.02 0.03 -0.03 0.01
FEAR30 0.03 0.00 0.04 -0.05 0.09 0.11 -0.02 0.10 0.00 -0.02 0.01 -0.07 0.66 0.07 0.06 -0.04 0.02 -0.03
FEAR31 0.04 0.03 -0.02 -0.03 0.04 0.01 0.01 -0.04 0.80 0.00 0.05 -0.02 0.09 -0.01 0.01 0.00 0.05 -0.03
SEPR32 0.07 -0.02 -0.03 0.00 -0.03 0.82 0.07 0.04 -0.01 -0.03 0.02 -0.02 0.01 0.02 -0.07 0.06 -0.02 -0.07
SEPR33 -0.08 0.03 0.02 0.03 0.01 0.78 -0.01 -0.03 0.05 0.07 -0.03 0.02 0.02 -0.02 0.03 0.01 0.03 -0.01
SEPR34 -0.03 0.00 0.00 0.06 0.03 0.77 -0.04 -0.04 -0.01 0.03 0.01 0.03 0.02 0.01 0.05 -0.06 -0.02 0.12
EXCITE35 0.13 0.04 -0.06 -0.03 -0.06 0.13 0.55 0.04 0.02 0.08 -0.07 0.15 -0.07 -0.02 0.06 0.13 -0.02 -0.01
EXCITE36 0.01 0.00 0.15 0.00 0.03 -0.04 0.62 0.00 -0.06 0.02 0.04 0.13 0.04 0.03 -0.04 0.07 0.01 -0.07
EXCITE37 0.01 0.05 -0.03 0.01 0.01 0.03 0.82 0.03 0.03 -0.01 0.02 -0.02 -0.02 -0.01 0.01 0.01 -0.11 0.00
EXCITE38 -0.06 -0.13 0.03 0.03 0.01 -0.02 0.78 0.00 0.02 0.00 0.07 -0.02 0.03 0.06 -0.01 -0.03 0.11 0.05
ATT39 0.05 0.03 0.06 0.02 -0.03 0.00 0.07 -0.02 -0.01 -0.01 -0.09 0.10 0.00 -0.10 0.06 0.74 -0.01 -0.01
ATT40 0.05 0.13 -0.10 -0.05 0.05 0.11 0.03 -0.06 -0.08 -0.06 0.19 -0.20 0.09 0.14 -0.04 0.49 -0.02 0.06
ATT41 -0.03 -0.01 0.03 -0.07 -0.05 0.08 -0.02 0.04 -0.03 0.01 0.09 0.10 0.02 0.05 0.16 0.40 0.16 0.13
PLAY43 -0.31 -0.01 0.50 -0.18 0.00 -0.03 0.07 -0.08 0.10 -0.06 -0.03 0.09 -0.01 0.06 0.05 -0.07 0.06 0.02
PLAY44 0.01 0.05 0.88 -0.05 -0.01 -0.01 0.04 0.09 -0.02 0.00 0.05 0.00 0.03 0.00 -0.08 0.02 -0.04 0.01
PLAY45 0.03 -0.02 0.86 0.03 -0.02 -0.03 -0.04 -0.06 0.05 -0.04 0.01 0.06 -0.04 0.04 0.03 0.05 0.00 -0.08
PLAY46 0.00 -0.08 0.79 0.12 0.00 0.04 0.02 0.00 -0.07 0.03 0.01 0.01 -0.04 -0.08 -0.01 -0.05 0.02 0.06
IMP47 0.02 -0.05 0.13 0.01 -0.02 -0.02 0.08 -0.01 0.05 0.13 0.61 0.09 0.02 0.00 0.09 -0.02 -0.04 0.11
IMP48 -0.08 -0.02 0.04 0.05 0.01 0.06 0.10 0.02 0.06 0.04 0.67 0.03 0.00 0.03 0.06 -0.01 0.02 -0.05
IMP49 0.01 0.11 0.00 0.04 -0.07 -0.10 0.12 0.02 0.04 0.21 0.35 0.00 0.12 -0.02 0.08 -0.05 0.00 0.12
MISC52 -0.06 0.01 -0.08 0.15 0.02 -0.05 0.01 -0.05 0.10 0.01 0.01 0.00 0.02 0.05 0.70 0.09 0.14 -0.07
MISC53 0.07 -0.01 0.00 -0.09 0.03 0.06 -0.01 0.06 -0.04 0.06 0.07 -0.06 0.01 0.04 0.74 -0.01 -0.13 0.07
MISC54 -0.03 0.11 -0.02 -0.01 0.64 -0.02 0.05 -0.02 0.02 0.03 0.02 -0.10 0.10 0.03 0.04 0.05 0.10 0.00
MISC55 -0.02 -0.04 0.04 0.00 0.84 0.04 0.00 0.04 0.01 0.03 -0.10 0.02 0.00 -0.04 -0.03 -0.01 0.06 0.11
MISC56 -0.03 0.02 -0.06 -0.01 0.83 -0.02 -0.01 -0.02 0.03 -0.02 0.08 0.04 -0.02 0.01 0.04 -0.03 -0.11 -0.04
MISC59 0.06 0.10 0.03 -0.10 0.01 0.28 -0.05 0.10 -0.07 -0.06 0.41 0.35 0.02 0.02 0.04 -0.03 -0.01 0.03
MISC60 -0.03 -0.03 -0.01 -0.04 0.04 -0.03 0.02 -0.10 0.03 0.05 0.04 0.75 0.06 0.09 -0.01 0.05 0.00 0.02
MISC61 0.03 0.01 0.10 0.04 -0.04 0.07 0.07 0.06 -0.06 -0.02 0.02 0.74 -0.04 -0.02 -0.06 0.01 0.00 0.02
MISC62 -0.03 0.18 -0.04 -0.05 -0.03 0.04 0.00 0.16 0.18 0.03 -0.01 0.10 0.00 0.57 0.04 0.01 -0.05 -0.02
MISC63 -0.02 0.12 -0.02 0.02 0.01 0.00 0.13 -0.06 -0.03 -0.02 0.04 0.06 0.06 0.65 0.12 -0.07 0.12 0.01
MISC64 -0.08 0.79 0.01 0.05 0.04 0.02 -0.02 0.01 0.01 -0.01 -0.07 0.02 0.03 0.03 0.03 0.03 -0.07 0.05
MISC65 -0.07 0.66 -0.02 0.02 -0.01 0.02 -0.02 0.06 0.05 0.06 0.01 -0.01 0.05 0.12 0.06 0.01 -0.02 0.07
MISC66 0.18 0.53 -0.04 -0.01 0.18 0.07 -0.04 -0.07 -0.01 0.05 0.12 -0.07 0.00 0.12 -0.03 0.04 0.10 -0.09
MISC67 0.08 0.55 -0.13 0.07 0.14 0.03 -0.04 0.01 0.06 -0.03 0.13 0.04 -0.02 0.01 -0.05 0.09 0.21 -0.17
MISC69 0.05 0.06 -0.05 0.01 0.14 0.03 0.03 0.00 0.05 -0.06 0.03 0.06 -0.02 -0.02 0.01 0.05 0.02 0.66
2.7.6.4.1 18 Factor Model Largest Loading Per Item
Largest Loading Per Item and Associated Factors, Sorted by Factor
largest maximum itemNames itemDescriptions
40 PA1 0.79 MISC64 When working, is easily distracted or preoccupied by odors/engages in persistent sniffing of ground or objects
41 PA1 0.66 MISC65 Has difficulty shifting attention away from interesting or distracting stimuli (e.g., other dogs, odor, people, small animals, etc.)
42 PA1 0.53 MISC66 Is distracted or nervous in new, unfamiliar environments, has difficulty maintaining focus on work
43 PA1 0.55 MISC67 Is slow to recover after being distracted, startled, or frightened/takes a long time to resume work
56 PA1 0.59 TRAIN07 Unfocused; is easily distracted by interesting sights, sounds or smells
30 PA10 0.70 MISC52 Begs persistently for food when people are eating
31 PA10 0.74 MISC53 Steals food
24 PA11 0.84 FEAR29 When having nails trimmed, or feet touched/handled
25 PA11 0.66 FEAR30 When groomed or bathed
17 PA12 0.55 FEAR22 In response to sudden or loud noises (e.g., gun fire, car backfire, road drills, objects being dropped, etc.)
20 PA12 0.49 FEAR25 In response to strange or unfamiliar objects on or near sidewalks or walkways (e.g., plastic trash bags, leaves, litter, flags flapping, etc.
36 PA13 0.75 MISC60 Playful, puppyish, boisterous
37 PA13 0.74 MISC61 Active, energetic, always on the go
9 PA14 0.74 ATT39 Displays a strong attachment for you or another familiar person
10 PA14 0.49 ATT40 Unwilling to leave your side, even when working, reluctant to work at a distance from you/the handler
11 PA14 0.40 ATT41 Tends to nudge or paw you (or others) for attention
27 PA15 0.61 IMP47 Impulsive; doesn’t seem to think before s/he acts
28 PA15 0.67 IMP48 Becomes frustrated/impatient in a wide range of situations
29 PA15 0.35 IMP49 Is difficult to interrupt or distract when doing things s/he wants to do
35 PA15 0.41 MISC59 Hyperactive, restless, has trouble settling down
44 PA16 0.66 MISC69 Chases/follows shadows, light spots, etc.
38 PA17 0.57 MISC62 Becomes highly excited and/or distracted when encountering unfamiliar dogs
39 PA17 0.65 MISC63 Becomes highly excited and/or distracted when encountering unfamiliar people
16 PA18 0.82 FEAR21 When an unfamiliar person approaches the dog when s/he is away from his/her normal home environment or kennel
18 PA18 0.76 FEAR23 When an unfamiliar person visits your home or approaches the dog when in his/her home kennel
19 PA18 0.90 FEAR24 When an unfamiliar person tries to touch or pet the dog
21 PA18 0.39 FEAR26 When examined/treated by a veterinarian
23 PA18 0.39 FEAR28 When first exposed to unfamiliar situations (e.g., novel environments, first visit to the veterinarian, etc.)
45 PA2 0.50 PLAY43 Eagerly engages in play with new/unfamiliar people
46 PA2 0.88 PLAY44 Highly toy focused; attention riveted on tug toy/balls when these are held by handler or other person
47 PA2 0.86 PLAY45 Eagerly initiates play sessions; brings objects/toys to you/the handler and retrieves them when thrown
48 PA2 0.79 PLAY46 Hunts persistently for thrown or hidden toys/objects, not easily distracted from this task
1 PA3 0.61 AGG09 Barks, growls, attempts to bite when approached directly by an unfamiliar person while being walked/exercised on a leash
2 PA3 0.83 AGG10 Barks, etc. when unfamiliar persons approach the dog when s/he is in his/her kennel
3 PA3 0.73 AGG14 When strangers walk past when the dog is in his/her home run or kennel
4 PA3 0.60 AGG15 When an unfamiliar person tries to touch or pet the dog
5 PA4 0.53 AGG16 When approached directly by an unfamiliar dog while being walked/exercised on a leash
6 PA4 0.56 AGG18 When barked, growled, or lunged at by another dog
7 PA4 0.84 AGG19 Towards other familiar dogs
8 PA4 0.71 AGG20 When approached while playing with/chewing a favorite toy, bone, object, etc. by another familiar dog
49 PA5 0.82 SEPR32 Restlessness/agitation/pacing when left alone
50 PA5 0.78 SEPR33 Barking or whining when left alone
51 PA5 0.77 SEPR34 Chewing/scratching at doors, floor, fencing, etc., when left alone
12 PA6 0.55 EXCITE35 Excitable when you first arrive home, or at the dog’s kennel, after a brief absence
13 PA6 0.62 EXCITE36 Excitable when playing with you or other familiar persons
14 PA6 0.82 EXCITE37 Excitable just before being taken out for a walk
15 PA6 0.78 EXCITE38 Excitable just before being taken out for work or training
32 PA7 0.64 MISC54 Reluctant to/nervous about crossing grates or other unfamiliar surfaces
33 PA7 0.84 MISC55 Reluctant to/nervous about crossing shiny or slippery floors
34 PA7 0.83 MISC56 Nervous or frightened when ascending or descending some types of stairs
52 PA8 0.46 TRAIN01 Is hard to recall when off the leash
53 PA8 0.62 TRAIN02 Is slow to obey a ‘sit’ command
54 PA8 0.74 TRAIN03 Is slow to obey a ‘stay’ command
55 PA8 0.47 TRAIN04 Has difficulty attending/listening to things you say or do
22 PA9 0.84 FEAR27 When approached directly by an unfamiliar dog
26 PA9 0.80 FEAR31 When barked, growled, or lunged at by an unfamiliar dog

2.7.7 18 Model Reliability Measures

PA18 PA1 PA2 PA3 PA7 PA5 PA6 PA4 PA9 PA8 PA15 PA13 PA11 PA17 PA10 PA14 PA12 PA16
alpha 0.797 0.805 0.812 0.76 0.718 0.735 0.787 0.752 0.803 0.693 0.692 0.71 0.616 0.64 0.622 0.494 0.447 1

omega is thought to be better measure of reliability than alpha, because it looks at the relationships between all the items rather than just the ones within each factor. It’s different than alpha because it assumes that there is a genral factor, g, that all items have in common. This is listed as the first row of the table below.

There are three omegas for each factor:

  • omega_total - total true score variance in factor
  • omega_hierarchical - variance attributable to general factor (g)
  • omega_subscale - variance attributable to the subscale

From EFAtools::OMEGA documentation:

Omegas refer to the correlation between a factor and a unit-weighted composite score and thus the true score variance in a unit-weighted composite based on the respective indicators. Omega total is the total true score variance in a composite. Omega hierarchical is the true score variance in a composite that is attributable to the general factor, and omega subscale is the true score variance in a composite attributable to all subscales / group factors (for the whole scale) or to the specific subscale / group factor (for subscale composites).

Interpretation of omega_subscale

no universal guideline

0.50 is minimum, 0.75 preferable (watkins books,rodrigues16, zimbarg2005

## Omega total, omega hierarchical, omega subscale, H index, explained common variance (ECV), and percent of uncontaminated correlations (PUC) for the general factor (top row) and omegas and H index for the group factors:
## 
##       tot  hier   sub     H   ECV   PUC
## g   0.952 0.607 0.165 0.896 0.236 0.954
## F1  0.816 0.398 0.418 0.674            
## F2  0.724 0.076 0.648 0.670            
## F3  0.751 0.207 0.544 0.629            
## F4  0.588 0.258 0.330 0.400            
## F5  0.686 0.001 0.685 0.712            
## F6  0.618 0.090 0.528 0.612            
## F7  0.567 0.188 0.380 0.569            
## F8  0.498 0.082 0.416 0.416            
## F9  0.630 0.254 0.376 0.454            
## F10 0.806 0.293 0.513 0.819            
## F11 0.827 0.067 0.760 0.874            
## F12 0.743 0.117 0.626 0.795            
## F13 0.741 0.155 0.586 0.770            
## F14 0.842 0.155 0.687 0.768            
## F15 0.748 0.024 0.724 0.816            
## F16 0.841 0.214 0.627 0.748            
## F17 0.662 0.180 0.482 0.648            
## F18 0.843 0.231 0.612 0.708

3 Model Comparisons

Model fit statistics for each number of factors. TLI = Tucker-Lewis index of factoring reliability
fits12 fits13 fits14 fits15 fits16 fits17 fits18
Bartlett test chi-square 32708.77 35060 34280.193 35265.38 37685.24 39253.54 38936
Bartlett test degrees of freedom 1035.00 1128 1128.000 1275.00 1431.00 1540.00 1540
Bartlett test p-value 0.00 0 0.000 0.00 0.00 0.00 0
KMO 0.85 1 0.849 0.85 0.85 0.85 1
Tucker-Lewis Index 0.76 1 0.760 0.75 0.72 0.72 1
Chi-square 4483.85 5318 4373.565 4709.92 5426.57 5699.96 5065
Degrees of freedom for chi-square 549.00 582 547.000 615.00 687.00 724.00 685
P-value for chi-square 0.00 0 0.000 0.00 0.00 0.00 0